Evanescent electromagnetic wave conversion methods II

ABSTRACT

Apparatus, methods, and systems provide conversion of evanescent electromagnetic waves to non-evanescent electromagnetic waves and/or conversion of non-evanescent electromagnetic waves to evanescent electromagnetic waves. In some approaches the conversion includes propagation of electromagnetic waves within an indefinite electromagnetic medium, and the indefinite medium may include an artificially-structured material such as a layered structure or other metamaterial.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is related to and claims the benefit of theearliest available effective filing date(s) from the following listedapplication(s) (the “Related Applications”) (e.g., claims earliestavailable priority dates for other than provisional patent applicationsor claims benefits under 35 USC §119(e) for provisional patentapplications, for any and all parent, grandparent, great-grandparent,etc. applications of the Related Application(s)). All subject matter ofthe Related Applications and of any and all parent, grandparent,great-grandparent, etc. applications of the Related Applications isincorporated herein by reference to the extent such subject matter isnot inconsistent herewith.

Related Applications

For purposes of the USPTO extra-statutory requirements, the presentapplication constitutes a continuation-in-part of U.S. patentapplication No. [NOT YET ASSIGNED], entitled EVANESCENT ELECTROMAGNETICWAVE CONVERSION APPARATUS I, naming Jeffrey A. Bowers, Roderick A. Hyde,Edward K. Y. Jung, John Brian Pendry, David Schurig, David R. Smith,Clarence T. Tegreene, Thomas Allan Weaver, Charles Whitmer, Lowell L.Wood, Jr. as inventors, filed Apr. 17, 2009, which is currentlyco-pending, or is an application of which a currently co-pendingapplication is entitled to the benefit of the filing date.

For purposes of the USPTO extra-statutory requirements, the presentapplication constitutes a continuation-in-part of U.S. patentapplication No. [NOT YET ASSIGNED], entitled EVANESCENT ELECTROMAGNETICWAVE CONVERSION APPARATUS III, naming Jeffrey A. Bowers, Roderick A.Hyde, Edward K. Y. Jung, John Brian Pendry, David Schurig, David R.Smith, Clarence T. Tegreene, Thomas Allan Weaver, Charles Whitmer,Lowell L. Wood, Jr. as inventors, filed Apr. 17, 2009, which iscurrently co-pending, or is an application of which a currentlyco-pending application is entitled to the benefit of the filing date.

The United States Patent Office (USPTO) has published a notice to theeffect that the USPTO's computer programs require that patent applicantsreference both a serial number and indicate whether an application is acontinuation or continuation-in-part. Stephen G. Kunin, Benefit ofPrior-Filed Application, USPTO Official Gazette Mar. 18, 2003, availableat http://www.uspto.gov/web/offices/com/sol/og/2003/week11/patbene.htm.The present Applicant Entity (hereinafter “Applicant”) has providedabove a specific reference to the application(s) from which priority isbeing claimed as recited by statute. Applicant understands that thestatute is unambiguous in its specific reference language and does notrequire either a serial number or any characterization, such as“continuation” or “continuation-in-part,” for claiming priority to U.S.patent applications. Notwithstanding the foregoing, Applicantunderstands that the USPTO's computer programs have certain data entryrequirements, and hence Applicant is designating the present applicationas a continuation-in-part of its parent applications as set forth above,but expressly points out that such designations are not to be construedin any way as any type of commentary and/or admission as to whether ornot the present application contains any new matter in addition to thematter of its parent application(s).

TECHNICAL FIELD

The application discloses apparatus and methods that may relate toelectromagnetic responses that include electromagnetic near-fieldlensing and/or conversion of evanescent electromagnetic waves tonon-evanescent electromagnetic waves and/or conversion of non-evanescentelectromagnetic waves to evanescent electromagnetic waves.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 depicts a conversion structure having first and second surfaceregions that are substantially planar and substantially parallel.

FIG. 2 depicts a layered structure as an exemplary implementation of theconversion structure of FIG. 1.

FIG. 3 depicts a layered structure as an exemplary implementation of theconversion structure of FIG. 1.

FIG. 4 depicts a conversion structure having a first surface region thatis substantially planar and a second surface region that issubstantially nonplanar.

FIG. 5 depicts a layered structure as an exemplary implementation of theconversion structure of FIG. 4.

FIG. 6 depicts a layered structure as an exemplary implementation of theconversion structure of FIG. 4.

FIG. 7 depicts a conversion structure having a first surface region thatis substantially nonplanar and a second surface region that issubstantially planar.

FIG. 8 depicts a layered structure as an exemplary implementation of theconversion structure of FIG. 7.

FIG. 9 depicts a layered structure as an exemplary implementation of theconversion structure of FIG. 7.

FIG. 10 depicts various conversion structures having first and secondsurface regions that are substantially nonplanar.

FIG. 11 depicts a layered structure as an exemplary implementation of aconversion structure as in FIG. 10.

FIG. 12 depicts a first process flow.

FIG. 13 depicts a second process flow reciprocal to the first processflow.

FIG. 14 depicts a system that includes an evanescent conversion unit.

DETAILED DESCRIPTION

In the following detailed description, reference is made to theaccompanying drawings, which form a part hereof. In the drawings,similar symbols typically identify similar components, unless contextdictates otherwise. The illustrative embodiments described in thedetailed description, drawings, and claims are not meant to be limiting.Other embodiments may be utilized, and other changes may be made,without departing from the spirit or scope of the subject matterpresented here.

Embodiments provide apparatus and methods for converting evanescentelectromagnetic waves to non-evanescent electromagnetic waves and/or forconverting non-evanescent electromagnetic waves to evanescentelectromagnetic waves. In general, an evanescent electromagnetic wave isan electromagnetic wave having an amplitude that decays exponentiallywith distance, e.g. having a wave vector that is at least partiallyimaginary. For example, the electric component of an electromagneticwave may have a 2D Fourier expansion given by

$\begin{matrix}{{E\left( {r,t} \right)} = {\sum\limits_{\sigma,k_{x},k_{y}}{{E_{\sigma}\left( {k_{x},k_{y}} \right)}{{\exp \left( {{\; k_{z}z} + {\; k_{x}x} + {\; k_{y}y} - {\; \omega \; t}} \right)}.}}}} & (1)\end{matrix}$

Supposing, for purposes of illustration, that the wave exists in amedium with refractive index n, the Fourier modes having k_(x) ²+k_(y)²<n²ω²/c² are propagating electromagnetic waves with real wavevectorcomponents k_(z)=+√{square root over (n²ω²c⁻²−k_(x) ²−k_(y) ²)}, whilethe Fourier modes having k_(x) ²+k_(y) ²>n²ω²/c² are evanescentelectromagnetic waves with imaginary wavevector components k_(z)=+i√{square root over (k_(x) ²+k_(y) ²−n²ω²c⁻²)}. The evanescentelectromagnetic waves decay exponentially with distance z. In aconventional far-field optics application, where z may represent, forexample, distance from an object plane of a conventional far-fieldoptical system, the evanescent waves do not substantially persist beyondan evanescent range μ˜1/|k_(z)|, corresponding to a near field of theobject plane (or a near field in the vicinity of an object to beimaged), while the propagating waves persist beyond the near field intothe far field to comprise a far-field image (e.g. on an image plane ofthe conventional far-field optical system). Thus, a conventionalfar-field optical system has a resolution limit Δ (sometimes referred toas a diffraction limit or an Abbe-Rayleigh limit) corresponding to amaximum transverse wavevector k_(max) for propagating waves:

$\begin{matrix}{{\left. \Delta \right.\sim\frac{2\; \pi}{k_{\max}}} = {\frac{2\; \pi \; c}{n\; \omega} = {\frac{c}{nv} = \frac{\lambda_{0}}{n}}}} & (2)\end{matrix}$

where λ₀ is the free-space wavelength corresponding to frequency v. Onthe other hand, embodiments disclosed herein provide apparatus andmethods that may exceed this resolution limit, by converting evanescentwaves to propagating waves (or vice versa) in an indefiniteelectromagnetic medium.

In general, an indefinite electromagnetic medium is an electromagneticmedium having electromagnetic parameters (e.g. permittivity and/orpermeability) that include indefinite tensor parameters. Throughout thisdisclosure, including the subsequent claims, the term “indefinite” istaken to have its algebraic meaning; thus, an indefinite tensor is atensor that is neither positive definite (having all positiveeigenvalues) nor negative definite (having all negative eigenvalues) butinstead has at least one positive eigenvalue and at least one negativeeigenvalue. Some exemplary indefinite media are described in D. R. Smithand D. Schurig, “Indefinite materials,” U.S. patent application Ser. No.10/525,191; D. R. Smith and D. Schurig, “Electromagnetic wavepropagation in media with indefinite permittivity and permeabilitytensors,” Phys. Rev. Lett. 90, 077405 (2003); and D. R. Smith and D.Schurig, “Sub-diffraction imaging with compensating bilayers,” New. J.Phys. 7, 162 (2005); each of which is herein incorporated by reference.

In some embodiments an indefinite medium is an electromagnetic mediumhaving an indefinite permeability. An example of an indefinitepermeability medium is a planar slab having a z-axis perpendicular tothe slab (with x- and y-axes parallel to the slab) and electromagneticparameters ε_(y), μ_(x), and μ_(z) satisfying the inequalities

ε_(y)μ_(x)>0, μ_(x)/μ_(z)<0   (3)

(thus, the permeability is indefinite, with either μ_(x)<0<μ_(z) orμ_(x)>0>μ_(z)). For TE-polarized (i.e. s-polarized) electromagneticwaves with an electric field directed along the y-axis, theseelectromagnetic parameters provide a hyperbolic dispersion relation

$\begin{matrix}{k_{z}^{2} = {{ɛ_{y}\mu_{x}\frac{\omega^{2}}{c^{2}}} - {\frac{\mu_{x}}{\mu_{z}}k_{x}^{2}}}} & (4)\end{matrix}$

that admits propagating electromagnetic waves (real k_(z)) with largetransverse wavevectors k_(x). Thus, if the planar slab adjoins a uniformrefractive medium with index of refraction n, an evanescent wave in theadjoining medium (e.g. as in equation (1), with k_(x)>nω/c) becomes apropagating wave in the indefinite medium (or, reciprocally, apropagating wave in the indefinite medium becomes an evanescent wave inthe adjoining medium). For sufficiently large k_(x) (i.e. substantiallywithin the asymptotic domain of the hyperbolic dispersion relation (4)),the propagating wave is characterized by group velocities that aresubstantially perpendicular to the asymptotes of equation (4), i.e. thepropagating wave is substantially conveyed along propagation directionsin the xz-plane that form an angle θ_(x)=tan⁻¹(|μ_(x)/μ_(z)|) withrespect to the z-axis (e.g. as depicted in FIG. 10 of the previouslycited U.S. patent application Ser. No. 10/525,191); moreover, forsufficiently small μ_(z) (i.e. |μ_(x)| substantially equal to zeroand/or substantially less than |μ_(z)|), the angle θ_(x) becomessubstantially equal to zero and the multiple propagating directionsdegenerate to a single propagation direction that substantiallycoincides with the z-axis (in this case the indefinite medium shall bereferred to as a degenerate indefinite medium). The planar slab mayalternately or additionally have electromagnetic parameters ε_(x) andμ_(y), satisfying the alternate or additional inequalities

ε_(x)μ_(y)>0, μ_(y)/μ_(z)<0,   (5)

providing another hyperbolic dispersion relation

$\begin{matrix}{k_{z}^{2} = {{ɛ_{x}\mu_{y}\frac{\omega^{2}}{c^{2}}} - {\frac{\mu_{y}}{\mu_{z}}k_{y}^{2}}}} & (6)\end{matrix}$

for TE-polarized electromagnetic waves with an electric field directedalong the x-axis. In this case, for sufficiently large k_(y) (i.e.substantially within the asymptotic domain of the hyperbolic dispersionrelation (6)), a propagating wave in the indefinite medium ischaracterized by group velocities that are substantially perpendicularto the asymptotes of equation (6), i.e. the propagating wave issubstantially conveyed along propagation directions in the yz-plane thatform an angle θ_(y)=tan⁻¹(|μ_(y)/μ_(z)|) with respect to the z-axis;moreover, for sufficiently small μ_(y) (i.e. |μ_(y)| substantially equalto zero and/or substantially less than |μ_(z)|), the angle θ_(y) becomessubstantially equal to zero and the multiple propagating directionsdegenerate to a single propagation direction that substantiallycoincides with the z-axis (another degenerate indefinite medium). Whenthe planar slab satisfies both inequalities (3) and (5), the indefinitemedium supports TE-polarized waves that substantially propagate (forsufficiently large transverse wavevectors k_(x) and/or k_(y)) alongpropagation directions that compose an elliptical cone having a coneaxis that coincides with the z-direction and half-angles θ_(x) andθ_(y), as above, with respect to the x- and y-axes, and in the casewhere ε_(x)=ε_(y) and μ_(x)=μ_(y), the planar slab is a uniaxial mediumthat provides the same hyperbolic dispersion for any TE-polarized waves,and the propagation directions for large transverse wavevectors composea circular cone with θ_(x)=θ_(y).

More generally, in some embodiments an indefinite permeability mediummay define an axial direction that corresponds to a first eigenvector ofthe indefinite permeability matrix, with first and second transversedirections that correspond to second and third eigenvectors of theindefinite permeability matrix, respectively. The parameters of theindefinite permeability matrix may vary with position within theindefinite permeability medium, and correspondingly the eigenvectors ofthe indefinite permeability matrix may also vary with position withinthe indefinite permeability medium. The disclosure of the precedingparagraph may encompass more general embodiments of an indefinitepermeability medium, in the following manner: the z-axis shall beunderstood to refer more generally to an axial direction that may varythroughout the indefinite medium, the x-axis shall be understood torefer more generally to a first transverse direction perpendicular tothe axial direction, and the y-axis shall be understood to refer moregenerally to a second transverse direction mutually perpendicular to theaxial direction and the first transverse direction. Thus, for example, auniaxial indefinite permeability medium may have a local axial parameterμ_(A) (corresponding to an axial direction that may vary with positionwithin the medium) and transverse parameters ε_(T1)=ε_(T2)=ε_(T),μ_(T1)=μ_(T2)=μ_(T) that satisfy the inequalities

ε_(T)μ_(T)>0, μ_(T)/μ_(A)<0,   (7)

providing a hyperbolic dispersion relation

$\begin{matrix}{{k_{A}^{2} = {{ɛ_{T}\mu_{T}\frac{\omega^{2}}{c^{2}}} - {\frac{\mu_{T}}{\mu_{A}}k_{T}^{2}}}},} & (8)\end{matrix}$

and this dispersion relation supports TE-polarized waves thatsubstantially propagate (for sufficiently large transverse wavevectorsk_(T)) along propagation directions that locally compose a circular conehaving a cone axis that coincides with the local axial direction with acone half-angle θ=tan⁻¹(|μ_(T)/μ_(A)|) (and where |μ_(T)|<<|μ_(A)|, themedium is a degenerate indefinite medium, wherein the cone ofpropagation directions degenerates to a single propagation directionthat substantially coincides with the local axial direction).

In some embodiments an indefinite medium is an electromagnetic mediumhaving an indefinite permittivity. An example of an indefinitepermittivity medium is a planar slab having a z-axis perpendicular tothe slab (with x- and y-axes parallel to the slab), and havingelectromagnetic parameters μ_(y), ε_(x), and ε_(z) satisfying theinequalities

μ_(y)ε_(x)>0, ε_(x)/μ_(z)<0   (9)

(thus, the permittivity is indefinite, with either ε_(x)<0<ε_(z) orε_(x)>0>ε_(z)). For TM-polarized (i.e. p-polarized) electromagneticwaves with a magnetic field directed along the y-axis, theseelectromagnetic parameters provide a hyperbolic dispersion relation

$\begin{matrix}{k_{z}^{2} = {{\mu_{y}ɛ_{x}\frac{\omega^{2}}{c^{2}}} - {\frac{ɛ_{x}}{ɛ_{z}}k_{x}^{2}}}} & (10)\end{matrix}$

that admits propagating electromagnetic waves (real k_(z)) with largetransverse wavevectors k_(x). Thus, if the planar slab adjoins a uniformrefractive medium with index of refraction n, an evanescent wave in theadjoining medium (e.g. as in equation (1), with k_(x)>nω/c) becomes apropagating wave in the indefinite medium (or, reciprocally, apropagating wave in the indefinite medium becomes an evanescent wave inthe adjoining medium). For sufficiently large k_(x) (i.e. substantiallywithin the asymptotic domain of the hyperbolic dispersion relation(10)), the propagating wave is characterized by group velocities thatare substantially perpendicular to the asymptotes of equation (10), i.e.the propagating wave is substantially conveyed along propagationdirections in the xz-plane that form an angle θ_(z)=tan⁻¹(|ε_(x)/ε_(z)|)with respect to the z-axis; moreover, for sufficiently small ε_(x) (i.e.|ε_(x)| substantially equal to zero and/or substantially less than|ε_(z)|), the angle θ_(x) becomes substantially equal to zero and themultiple propagating directions degenerate to a single propagationdirection that substantially coincides with the z-axis (in this case theindefinite medium shall be referred to as a degenerate indefinitemedium). The planar slab may alternately or additionally haveelectromagnetic parameters μ_(x) and ε_(y), satisfying the alternativeor additional inequalities

μ_(x)ε_(y)>0, ε_(y)/ε_(z)<0,   (11)

providing another hyperbolic dispersion relation

$\begin{matrix}{k_{z}^{2} = {{\mu_{x}ɛ_{y}\frac{\omega^{2}}{c^{2}}} - {\frac{ɛ_{y}}{ɛ_{z}}k_{y}^{2}}}} & (12)\end{matrix}$

for TM-polarized electromagnetic waves with a magnetic field directedalong the x-axis. In this case, for sufficiently large k_(y) (i.e.substantially within the asymptotic domain of the hyperbolic dispersionrelation (12)), a propagating wave in the indefinite medium ischaracterized by group velocities that are substantially perpendicularto the asymptotes of equation (12), i.e. the propagating wave issubstantially conveyed along propagation directions in the yz-plane thatform an angle θ_(y)=tan⁻¹(|ε_(y)/ε_(z)|) with respect to the z-axis;moreover, for sufficiently small ε_(y) (i.e. |ε_(y)| substantially equalto zero and/or substantially less than |ε_(z)|), the angle θ_(y) becomessubstantially equal to zero and the multiple propagation directionsdegenerate to a single propagation direction that substantiallycoincides with the z-axis (another degenerate indefinite medium). Whenthe planar slab satisfies both inequalities (9) and (11), the indefinitemedium supports TM-polarized waves that substantially propagate (forsufficiently large transverse wavevectors k_(x) and/or k_(y)) alongpropagation directions that compose an elliptical cone having a coneaxis that coincides with the z-direction and half-angles θ_(x) andθ_(y), as above, with respect to the x- and y-axes, and in the casewhere ε_(x)=ε_(y) and μ_(x)=μ_(y), the planar slab is a uniaxial mediumthat provides the same hyperbolic dispersion for any TM-polarized waves,and the propagation directions for large transverse wavevectors composea circular cone with θ_(x)=θ_(y).

More generally, in some embodiments an indefinite permittivity mediummay define an axial direction that corresponds to a first eigenvector ofthe indefinite permittivity matrix, with first and second transversedirections that correspond to second and third eigenvectors of theindefinite permittivity matrix, respectively. The parameters of theindefinite permittivity matrix may vary with position within theindefinite permittivity medium, and correspondingly the eigenvectors ofthe indefinite permittivity matrix may also vary with position withinthe indefinite permittivity medium. The disclosure of the precedingparagraph may encompass more general embodiments of an indefinitepermittivity medium, in the following manner: the z-axis shall beunderstood to refer more generally to an axial direction that may varythroughout the indefinite medium, the x-axis shall be understood torefer more generally to a first transverse direction perpendicular tothe axial direction, and the y-axis shall be understood to refer moregenerally to a second transverse direction mutually perpendicular to theaxial direction and the first transverse direction. Thus, for example, auniaxial indefinite permittivity medium may have a local axial parameterε_(A) (corresponding to an axial direction that may vary with positionwithin the medium) and transverse parameters ε_(T1)=E_(T2)=ε_(T),μ_(T1)=μ_(T2)=μ_(T) that satisfy the inequalities

ε_(T)μ_(T)>0, ε_(T)/ε_(A)<0,   (13)

providing a hyperbolic dispersion relation

$\begin{matrix}{{k_{A}^{2} = {{ɛ_{T}\mu_{T}\frac{\omega^{2}}{c^{2}}} - {\frac{ɛ_{T}}{ɛ_{A}}k_{T}^{2}}}},} & (14)\end{matrix}$

and this dispersion relation supports TM-polarized waves thatsubstantially propagate (for sufficiently large transverse wavevectorsk_(T)) along propagation directions that locally compose a circular conehaving a cone axis that coincides with the local axial direction with acone half-angle θ=tan⁻¹(|ε_(T)/ε_(A)|) (and where |ε_(T)|<<|ε_(A)|, themedium is a degenerate indefinite medium, wherein the cone ofpropagation directions degenerates to a single propagation directionthat substantially coincides with the local axial direction).

In some embodiments an indefinite medium is an electromagnetic mediumthat is “doubly indefinite,” i.e. having both an indefinite permittivityand an indefinite permeability. An example of a doubly indefinite mediumis a planar slab defining a z-axis perpendicular to the slab (with x-and y-axes parallel to the slab), and having electromagnetic parameterssatisfying one or both of equations (3) and (5) (providing indefinitepermeability) and one or both of equations (9) and (11) (providingindefinite permittivity). The doubly-indefinite planar slab provides ahyperbolic dispersion relation for at least one TE-polarized wave (as inequations (4) and/or (6)) and further provides a hyperbolic dispersionrelation for at least one TM-polarized wave (as in equations (10) and(12)), with wave propagation features as discussed in the precedingparagraphs containing the equations that are referenced here.

In some embodiments a doubly-indefinite medium may have an indefinitepermittivity matrix and an indefinite permeability matrix that aresubstantially simultaneously diagonalizable, and the doubly-indefinitemedium defines an axial direction that corresponds to a first commoneigenvector of the indefinite matrices, with first and second transversedirections that correspond to second and third common eigenvectors ofthe indefinite matrices, respectively. As in the preceding examples, theparameters of the indefinite matrices may vary with position within thedoubly-indefinite medium, and correspondingly the common eigenvectors ofthe indefinite matrices may also vary with position within thedoubly-indefinite medium. The disclosure of the preceding paragraph mayencompass more general embodiments of a doubly-indefinite medium, in thefollowing manner: the z-axis shall be understood to refer more generallyto an axial direction that may vary throughout the doubly-indefinitemedium, the x-axis shall be understood to refer more generally to afirst transverse direction perpendicular to the axial direction, and they-axis shall be understood to refer more generally to a secondtransverse direction mutually perpendicular to the axial direction andthe first transverse direction. Thus, for example, a uniaxialdoubly-indefinite medium may have local axial parameters ε_(A), μ_(A)(corresponding to an axial direction that may vary with position withinthe medium) and transverse parameters ε_(T1)=ε_(T2)=ε_(T),μ_(T1)=μ_(T2)=μ_(T) that satisfy the inequalities (7) and (13),providing hyperbolic dispersion relations (8) and (14), and thesedispersion relations respectively support TE- and TM-polarized waveswithin the doubly-indefinite medium, as discussed in the precedingparagraphs containing the equations that are referenced here.

Some embodiments provide an indefinite medium that is a transformationmedium, i.e. an electromagnetic medium having properties that may becharacterized according to transformation optics. Transformation opticsis an emerging field of electromagnetic engineering, and transformationoptics devices include structures that influence electromagnetic waves,where the influencing imitates the bending of electromagnetic waves in acurved coordinate space (a “transformation” of a flat coordinate space),e.g. as described in A. J. Ward and J. B. Pendry, “Refraction andgeometry in Maxwell's equations,” J. Mod. Optics 43, 773 (1996), J. B.Pendry and S. A. Ramakrishna, “Focusing light using negativerefraction,” J. Phys. [Cond. Matt.] 15, 6345 (2003), D. Schurig et al,“Calculation of material properties and ray tracing in transformationmedia,” Optics Express 14, 9794 (2006) (“D. Schurig et al (1)”), and inU. Leonhardt and T. G. Philbin, “General relativity in electricalengineering,” New J. Phys. 8, 247 (2006), each of which is hereinincorporated by reference. The use of the term “optics” does not implyany limitation with regards to wavelength; a transformation opticsdevice may be operable in wavelength bands that range from radiowavelengths to visible wavelengths and beyond.

A first exemplary transformation optics device is the electromagneticcloak that was described, simulated, and implemented, respectively, inJ. B. Pendry et al, “Controlling electromagnetic waves,” Science 312,1780 (2006); S. A. Cummer et al, “Full-wave simulations ofelectromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006);and D. Schurig et al, “Metamaterial electromagnetic cloak at microwavefrequencies,” Science 314, 977 (2006) (“D. Schurig et al (2)”); each ofwhich is herein incorporated by reference. See also J. Pendry et al,“Electromagnetic cloaking method,” U.S. patent application Ser. No.11/459,728, herein incorporated by reference. For the electromagneticcloak, the curved coordinate space is a transformation of a flat spacethat has been punctured and stretched to create a hole (the cloakedregion), and this transformation corresponds to a set of constitutiveparameters (electric permittivity and magnetic permeability) for atransformation medium wherein electromagnetic waves are refracted aroundthe hole in imitation of the curved coordinate space.

A second exemplary transformation optics device is illustrated byembodiments of the electromagnetic compression structure described in J.B. Pendry, D. Schurig, and D. R. Smith, “Electromagnetic compressionapparatus, methods, and systems,” U.S. patent application Ser. No.11/982,353; and in J. B. Pendry, D. Schurig, and D. R. Smith,“Electromagnetic compression apparatus, methods, and systems,” U.S.patent application Ser. No. 12/069,170; each of which is hereinincorporated by reference. In embodiments described therein, anelectromagnetic compression structure includes a transformation mediumwith constitutive parameters corresponding to a coordinatetransformation that compresses a region of space intermediate first andsecond spatial locations, the effective spatial compression beingapplied along an axis joining the first and second spatial locations.The electromagnetic compression structure thereby provides an effectiveelectromagnetic distance between the first and second spatial locationsgreater than a physical distance between the first and second spatiallocations.

A third exemplary transformation optics device is illustrated byembodiments of the electromagnetic cloaking and/or translation structuredescribed in J. T. Kare, “Electromagnetic cloaking apparatus, methods,and systems,” U.S. patent application Ser. No. 12/074,247; and in J. T.Kare, “Electromagnetic cloaking apparatus, methods, and systems,” U.S.patent application Ser. No. 12/074,248; each of which is hereinincorporated by reference. In embodiments described therein, anelectromagnetic translation structure includes a transformation mediumthat provides an apparent location of an electromagnetic transducerdifferent then an actual location of the electromagnetic transducer,where the transformation medium has constitutive parameterscorresponding to a coordinate transformation that maps the actuallocation to the apparent location. Alternatively or additionally,embodiments include an electromagnetic cloaking structure operable todivert electromagnetic radiation around an obstruction in a field ofregard of the transducer (and the obstruction can be anothertransducer).

A fourth exemplary transformation optics device is illustrated byembodiments of the various focusing and/or focus-adjusting structuresdescribed in J. A. Bowers et al, “Focusing and sensing apparatus,methods, and systems,” U.S. patent application Ser. No. 12/156,443; J.A. Bowers et al, “Emitting and focusing apparatus, methods, andsystems,” U.S. patent application Ser. No. 12/214,534; J. A. Bowers etal, “Negatively-refractive focusing and sensing apparatus, methods, andsystems,” U.S. patent application Ser. No. 12/220,705; J. A. Bowers etal, “Emitting and negatively-refractive focusing apparatus, methods, andsystems,” U.S. patent application Ser. No. 12/220,703; J. A. Bowers etal, “Negatively-refractive focusing and sensing apparatus, methods, andsystems,” U.S. patent application Ser. No. 12/228,140; and J. A. Bowerset al, “Emitting and negatively-refractive focusing apparatus, methods,and systems,” U.S. patent application Ser. No. 12/228,153; each of whichis herein incorporated by reference. In embodiments described therein, afocusing and/or focusing-structure includes a transformation medium thatprovides an extended depth of focus/field greater than a nominal depthof focus/field, or an interior focus/field region with an axialmagnification that is substantially greater than or less than one.

Additional exemplary transformation optics devices are described in D.Schurig et al, “Transformation-designed optical elements,” Opt. Exp. 15,14772 (2007); M. Rahm et al, “Optical design of reflectionless complexmedia by finite embedded coordinate transformations,” Phys. Rev. Lett.100, 063903 (2008); and A. Kildishev and V. Shalaev, “Engineering spacefor light via transformation optics,” Opt. Lett. 33, 43 (2008); each ofwhich is herein incorporated by reference.

In general, for a selected coordinate transformation, a transformationmedium can be identified wherein electromagnetic fields evolve as in acurved coordinate space corresponding to the selected coordinatetransformation. Constitutive parameters of the transformation medium canbe obtained from the equations:

ε ^(i′j′)=[det(Λ)]⁻¹Λ_(i) ^(i′)Λ_(j) ^(j′)ε^(ij)   (15)

μ ^(i′j′)=[det(Λ)]⁻¹Λ_(i) ^(i′)Λ_(j) ^(j′)μ^(ij)   (16)

where ε and μ are the permittivity and permeability tensors of thetransformation medium, ε and μ are the permittivity and permeabilitytensors of the original medium in the untransformed coordinate space,and

$\begin{matrix}{\Lambda_{i}^{i^{\prime}} = \frac{\partial x^{i^{\prime}}}{\partial x^{i}}} & (17)\end{matrix}$

is the Jacobian matrix corresponding to the coordinate transformation.In some applications, the coordinate transformation is a one-to-onemapping of locations in the untransformed coordinate space to locationsin the transformed coordinate space, and in other applications thecoordinate transformation is a one-to-many mapping of locations in theuntransformed coordinate space to locations in the transformedcoordinate space. Some coordinate transformations, such as one-to-manymappings, may correspond to a transformation medium having a negativeindex of refraction. In some applications, the transformation medium isan indefinite medium, i.e. an electromagnetic medium having anindefinite permittivity and/or an indefinite permeability (thesetransformation media may be referred to as “indefinite transformationmedia”). For example, in equations (15) and (16), if the originalpermittivity matrix ε is indefinite, then the transformed permittivitymatrix ε is also indefinite; and/or if the original permeability matrixμ is indefinite, then the transformed permeability matrix μ is alsoindefinite. In some applications, only selected matrix elements of thepermittivity and permeability tensors need satisfy equations (15) and(16), e.g. where the transformation optics response is for a selectedpolarization only. In other applications, a first set of permittivityand permeability matrix elements satisfy equations (15) and (16) with afirst Jacobian Λ, corresponding to a first transformation opticsresponse for a first polarization of electromagnetic waves, and a secondset of permittivity and permeability matrix elements, orthogonal (orotherwise complementary) to the first set of matrix elements, satisfyequations (15) and (16) with a second Jacobian Λ′, corresponding to asecond transformation optics response for a second polarization ofelectromagnetic waves. In yet other applications, reduced parameters areused that may not satisfy equations (15) and (16), but preserve productsof selected elements in (15) and selected elements in (16), thuspreserving dispersion relations inside the transformation medium (see,for example, D. Schurig et al (2), supra, and W. Cai et al, “Opticalcloaking with metamaterials,” Nature Photonics 1, 224 (2007), hereinincorporated by reference). Reduced parameters can be used, for example,to substitute a magnetic response for an electric response, or viceversa. While reduced parameters preserve dispersion relations inside thetransformation medium (so that the ray or wave trajectories inside thetransformation medium are unchanged from those of equations (15) and(16)), they may not preserve impedance characteristics of thetransformation medium, so that rays or waves incident upon a boundary orinterface of the transformation medium may sustain reflections (whereasin general a transformation medium according to equations (15) and (16)is substantially nonreflective or sustains the reflectioncharacteristics of the original medium in the untransformed coordinatespace). The reflective or scattering characteristics of a transformationmedium with reduced parameters can be substantially reduced oreliminated (modulo any reflection characteristics of the original mediumin the untransformed coordinate space) by a suitable choice ofcoordinate transformation, e.g. by selecting a coordinate transformationfor which the corresponding Jacobian Λ (or a subset of elements thereof)is continuous or substantially continuous at a boundary or interface ofthe transformation medium (see, for example, W. Cai et al, “Nonmagneticcloak with minimized scattering,” Appl. Phys. Lett. 91, 111105 (2007),herein incorporated by reference).

Embodiments of an indefinite medium and/or a transformation medium(including embodiments of indefinite transformation media) can berealized using artificially-structured materials. Generally speaking,the electromagnetic properties of artificially-structured materialsderive from their structural configurations, rather than or in additionto their material composition.

In some embodiments, the artificially-structured materials are photoniccrystals. Some exemplary photonic crystals are described in J. D.Joannopoulos et al, Photonic Crystals. Molding the Flow of Light, 2^(nd)Edition, Princeton Univ. Press, 2008, herein incorporated by reference.In a photonic crystals, photonic dispersion relations and/or photonicband gaps are engineered by imposing a spatially-varying pattern on anelectromagnetic material (e.g. a conducting, magnetic, or dielectricmaterial) or a combination of electromagnetic materials. The photonicdispersion relations translate to effective constitutive parameters(e.g. permittivity, permeability, index of refraction) for the photoniccrystal. The spatially-varying pattern is typically periodic,quasi-periodic, or colloidal periodic, with a length scale comparable toan operating wavelength of the photonic crystal.

In other embodiments, the artificially-structured materials aremetamaterials. Some exemplary metamaterials are described in R. A. Hydeet al, “Variable metamaterial apparatus,” U.S. patent application Ser.No. 11/355,493; D. Smith et al, “Metamaterials,” InternationalApplication No. PCT/US2005/026052; D. Smith et al, “Metamaterials andnegative refractive index,” Science 305, 788 (2004); D. Smith et al,“Indefinite materials,” U.S. patent application Ser. No. 10/525,191; C.Caloz and T. Itoh, Electromagnetic Metamaterials. Transmission LineTheory and Microwave Applications, Wiley-Interscience, 2006; N. Enghetaand R. W. Ziolkowski, eds., Metamaterials: Physics and EngineeringExplorations, Wiley-Interscience, 2006; and A. K. Sarychev and V. M.Shalaev, Electrodynamics of Metamaterials, World Scientific, 2007; eachof which is herein incorporated by reference.

Metamaterials generally feature subwavelength elements, i.e. structuralelements with portions having electromagnetic length scales smaller thanan operating wavelength of the metamaterial, and the subwavelengthelements have a collective response to electromagnetic radiation thatcorresponds to an effective continuous medium response, characterized byan effective permittivity, an effective permeability, an effectivemagnetoelectric coefficient, or any combination thereof. For example,the electromagnetic radiation may induce charges and/or currents in thesubwavelength elements, whereby the subwavelength elements acquirenonzero electric and/or magnetic dipole moments. Where the electriccomponent of the electromagnetic radiation induces electric dipolemoments, the metamaterial has an effective permittivity; where themagnetic component of the electromagnetic radiation induces magneticdipole moments, the metamaterial has an effective permeability; andwhere the electric (magnetic) component induces magnetic (electric)dipole moments (as in a chiral metamaterial), the metamaterial has aneffective magnetoelectric coefficient. Some metamaterials provide anartificial magnetic response; for example, split-ring resonators(SRRs)—or other LC or plasmonic resonators—built from nonmagneticconductors can exhibit an effective magnetic permeability (cf. J. B.Pendry et al, “Magnetism from conductors and enhanced nonlinearphenomena,” IEEE Trans. Micro. Theo. Tech. 47, 2075 (1999), hereinincorporated by reference). Some metamaterials have “hybrid”electromagnetic properties that emerge partially from structuralcharacteristics of the metamaterial, and partially from intrinsicproperties of the constituent materials. For example, G. Dewar, “A thinwire array and magnetic host structure with n<0,” J. Appl. Phys. 97,10Q101 (2005), herein incorporated by reference, describes ametamaterial consisting of a wire array (exhibiting a negativepermeability as a consequence of its structure) embedded in anonconducting ferrimagnetic host medium (exhibiting an intrinsicnegative permeability). Metamaterials can be designed and fabricated toexhibit selected permittivities, permeabilities, and/or magnetoelectriccoefficients that depend upon material properties of the constituentmaterials as well as shapes, chiralities, configurations, positions,orientations, and couplings between the subwavelength elements. Theselected permittivities, permeabilities, and/or magnetoelectriccoefficients can be positive or negative, complex (having loss or gain),anisotropic (including tensor-indefinite), variable in space (as in agradient index lens), variable in time (e.g. in response to an externalor feedback signal), variable in frequency (e.g. in the vicinity of aresonant frequency of the metamaterial), or any combination thereof. Theselected electromagnetic properties can be provided at wavelengths thatrange from radio wavelengths to infrared/visible wavelengths; the latterwavelengths are attainable, e.g., with nanostructured materials such asnanorod pairs or nano-fishnet structures (cf. S. Linden et al, “Photonicmetamaterials: Magnetism at optical frequencies,” IEEE J. Select. Top.Quant. Elect. 12, 1097 (2006) and V. Shalaev, “Optical negative-indexmetamaterials,” Nature Photonics 1, 41 (2007), both herein incorporatedby reference). An example of a three-dimensional metamaterial at opticalfrequencies, an elongated-split-ring “woodpile” structure, is describedin M. S. Rill et al, “Photonic metamaterials by direct laser writing andsilver chemical vapour deposition,” Nature Materials advance onlinepublication, May 11, 2008, (doi: 10.1038/nmat2197).

While many exemplary metamaterials are described as including discreteelements, some implementations of metamaterials may include non-discreteelements or structures. For example, a metamaterial may include elementscomprised of sub-elements, where the sub-elements are discretestructures (such as split-ring resonators, etc.), or the metamaterialmay include elements that are inclusions, exclusions, or othervariations along some continuous structure (e.g. etchings on asubstrate). The metamaterial may include extended structures havingdistributed electromagnetic responses (such as distributed inductiveresponses, distributed capacitive responses, and distributedinductive-capacitive responses). Examples include structures consistingof loaded and/or interconnected transmission lines (such as microstripsand striplines), artificial ground plane structures (such as artificialperfect magnetic conductor (PMC) surfaces and electromagnetic band gap(EGB) surfaces), and interconnected/extended nanostructures(nano-fishnets, elongated SRR woodpiles, etc.).

In some embodiments a metamaterial may include a layered structure. Forexample, embodiments may provide a structure having a succession ofadjacent layers, where the layers have a corresponding succession ofmaterial properties that include electromagnetic properties (such aspermittivity and/or permeability). The succession of adjacent layers maybe an alternating or repeating succession of adjacent layers, e.g. astack of layers of a first type interleaved with layers of a secondtype, or a stack that repeats a sequence of three or more types oflayers. When the layers are sufficiently thin (e.g. having thicknessessmaller than an operating wavelength of the metamaterial), the layeredstructure may be characterized as an effective continuous medium havingeffective constitutive parameters that relate to the electromagneticproperties of the individual layers. For example, consider a planarstack of layers of a first material (of thickness d₁, and havinghomogeneous and isotropic electromagnetic parameters ε₁, μ₁) interleavedwith layers of a second material (of thickness d₂, and havinghomogeneous and isotropic electromagnetic parameters ε₂, μ₁); then thelayered structure may be characterized as an effective continuous mediumhaving (effective) anisotropic constitutive parameters

$\begin{matrix}{{ɛ_{x} = {ɛ_{y} = \frac{ɛ_{1} + {\eta \; ɛ_{2}}}{1 + \eta}}},} & (18) \\{{\frac{1}{ɛ_{z}} = {\frac{1}{1 + \eta}\left( {\frac{1}{ɛ_{1}} + \frac{\eta}{ɛ_{2}}} \right)}},} & (19) \\{{\mu_{x} = {\mu_{y} = \frac{\mu_{1} + {\eta\mu}_{2}}{1 + \eta}}},} & (20) \\{\frac{1}{\mu_{z}} = {\frac{1}{1 + \eta}\left( {\frac{1}{\mu_{1}} + \frac{\eta}{\mu_{2}}} \right)}} & (21)\end{matrix}$

where η=d₂/d₁ is the ratio of the two layer thicknesses, z is thedirection normal to the layers, and x, y are the directions parallel tothe layers. When the two materials comprising the interleaved structurehave electromagnetic parameters that are oppositely-signed, theconstitutive parameters (18)-(21) may correspond to an indefinitemedium. For example, when the first material is a material having apermittivity ε₁<0 and the second material is a material having apermittivity ε₂>0, the ratio η may be selected to provide an indefinitepermittivity matrix according to equations (18)-(19) (moreover, for ηsubstantially equal to |ε₁/ε₂|, the indefinite permittivity medium issubstantially a degenerate indefinite permittivity medium). Alternatelyor additionally, when the first material is a material having apermeability μ₁<0 and the second material is a material having apermeability μ₂>0, the ratio η may be selected to provide an indefinitepermeability matrix according to equations (20)-(21) (moreover, for ηsubstantially equal to |μ₁/μ₂|, the indefinite permeability medium issubstantially a degenerate indefinite permeability medium).

Exemplary planar stacks of alternating materials, providing an effectivecontinuous medium having an indefinite permittivity matrix, include analternating silver/silica layered system described in B. Wood et al,“Directed subwavelength imaging using a layered medal-dielectricsystem,” Phys. Rev. B 74, 115116 (2006), and an alternatingdoped/undoped semiconductor layered system described in A. J.

Hoffman, “Negative refraction in semiconductor metamaterials,” NatureMaterials 6, 946 (2007), each of which is herein incorporated byreference. More generally, materials having a positive permittivityinclude but are not limited to: semiconductors (e.g. at frequencieshigher than a plasma frequency of the semiconductor) and insulators suchas dielectric crystals (e.g. silicon oxide, aluminum oxide, calciumfluoride, magnesium fluoride), glasses, ceramics, and polymers (e.g.photoresist, PMMA). Generally these materials have a positivepermeability as well (which may be substantially equal to unity if thematerial is substantially nonmagnetic). In some embodiments a positivepermittivity material is a gain medium, which may be pumped, forexample, to reduce or overcome other losses such as ohmic losses (cf. anexemplary alternating silver/gain layered system described in S.Ramakrishna and J.

B. Pendry, “Removal of absorption and increase in resolution in anear-field lens via optical gain,” Phys. Rev. B 67, 201101 (R) (2003),herein incorporated by reference). Examples of gain media includesemiconductor laser materials (e.g. GaN, AlGaAs), doped insulator lasermaterials (e.g. rare-earth doped crystals, glasses, or ceramics), andRaman gain materials. Materials having a negative permeability includebut are not limited to: ferrites, magnetic garnets or spinels,artificial ferrites, and other ferromagnetic or ferrimagnetic materials(e.g. at frequencies above a ferromagnetic or ferrimagnetic resonancefrequency of the material; cf. F. J. Rachford, “Tunable negativerefractive index composite,” U.S. patent application Ser. No.11/279/460, herein incorporated by reference). Materials having anegative permittivity include but are not limited to: metals (e.g. atfrequencies less than a plasma frequency of the metal) including thenoble metals (Cu, Au, Ag); semiconductors (e.g. at frequencies less thana plasma frequency of the semiconductor); and polar crystals (e.g. SiC,LiTaO₃, LiF, ZnSe) at frequencies within a restrahlen band of the polarcrystal (cf. G. Schvets, “Photonic approach to making a material with anegative index of refraction,” Phys. Rev. B 67, 035109 (2003) and T.Tauber et al, “Near-field microscopy through a SiC superlens,” Science313, 1595 (2006), each of which is herein incorporated by reference).For applications involving semiconductors, the plasma frequency (whichmay be regarded as a frequency at which the semiconductor permittivitychanges sign) is related to the density of free carriers within thesemiconductor, and this free carrier density may be controlled invarious ways (e.g. by chemical doping, photodoping, temperature change,carrier injection, etc.). Thus, for example, a layered system comprisinginterleaved layers of a first semiconductor material having a firstplasma frequency and a second semiconductor material having a secondplasma frequency may provide an indefinite permittivity (per equations(18)-(19)) in a window of frequencies intermediate the first plasmafrequency and the second plasma frequency, and this window may becontrolled, e.g., by differently doping the first and secondsemiconductor materials.

In some applications a layered structure includes a succession ofadjacent layers that are substantially nonplanar. The precedingexemplary layered structure—consisting of successive planar layers, eachlayer having a layer normal direction (the z direction) that is constantalong the transverse extent of the layer and a layer thickness that isconstant along the transverse extent of the layer—may be extended to anonplanar layered structure, consisting of successive nonplanar layers,each layer having a layer normal direction that is non-constant alongthe transverse extent of the layer and, optionally, a layer thicknessthat is non-constant along the transverse extent of the layer. Someexamples of cylindrical and/or spherical layered structures aredescribed in A. Salandrino and N. Engheta, “Far-field subdiffractionoptical microscopy using metamaterial crystals: Theory and simulations,”Phys. Rev. B 74, 075103 (2006); Z. Jacob et al, “Optical hyperlens:Far-field imaging beyond the diffraction limit,” Opt. Exp. 14, 8247(2006); Z. Liu et al, “Far field optical hyperlens magnifyingsub-diffraction-limited objects,” Science 315, 1686 (2007); and H. Lee,“Development of optical hyperlens for imaging below the diffractionlimit,” Opt. Exp. 15, 15886 (2007); each of which is herein incorporatedby reference. More generally, for an alternating nonplanar layeredstructure, supposing that the layers have curvature radii substantiallyless than their respective thicknesses, and transverse layer thicknessgradients substantially less than one, the nonplanar layered structuremay be characterized as an effective continuous medium having(effective) anisotropic constitutive parameters as in equations(18)-(21), where the z direction is replaced by a layer normal directionthat may vary with position within the layered structure, the xdirection is replaced by a first transverse direction perpendicular tothe layer normal direction, the y direction is replaced by a secondtransverse direction mutually perpendicular to the layer normaldirection and the first transverse direction, and the layer thicknessratio η=d₂/d₁ is a ratio of local layer thicknesses d₁ and d₂ that mayvary with position throughout the layered structure (so the ratio η mayvary with position as well). The nonplanar layered structure may thusprovide an indefinite medium having a spatially-varying axial directionthat corresponds to the layer normal direction. Suppose, for example,that the spatially-varying axial direction of an indefinite medium isgiven by a vector field u_(A)(r) that is equal to or parallel to aconservative vector field, i.e.

u_(A)∝∇Φ  (22)

for a scalar potential function Φ; then the indefinite medium may beprovided by a nonplanar layered structure where the interfaces ofadjacent layers correspond to equipotential surfaces of the function Φ.

Nonplanar layered structures may be fabricated by various methods thatare known to those of skill in the art. In a first example, J. A. Folta,“Dynamic mask for producing uniform or graded-thickness thin films,”U.S. Pat. No. 7,062,348 (herein incorporated by reference), describesvapor deposition systems that utilize a moving mask, where the velocityand position of the moving mask may be computer controlled to preciselytailor the thickness distributions of deposited films. In a secondexample, Tzu-Yu Wang, “Graded thickness optical element and method ofmanufacture therefor,” U.S. Pat. No. 6,606,199 (herein incorporated byreference), describes methods for depositing graded thickness layersthrough apertures in a masking layer.

With reference now to FIG. 1, an illustrative embodiment is depictedthat includes a conversion structure 100 with indefinite electromagneticparameters, the conversion structure having a first surface region 111and a second surface region 112, the first surface region and the secondsurface region being substantially planar and substantially parallel.This and other drawings, unless context dictates otherwise, canrepresent a planar view of a three-dimensional embodiment, or atwo-dimensional embodiment (e.g. in FIG. 1 where the conversionstructure is placed inside a metallic or dielectric slab waveguideoriented normal to the page). The conversion structure is responsive toan evanescent electromagnetic wave (depicted schematically asexponential tails 120) at the first surface region to convey apropagating electromagnetic wave (depicted schematically as dashed rays125) from the first surface region to the second surface region, and toprovide a non-evanescent electromagnetic wave (depicted schematically asthe wavy rays 130) at the second surface region. In some embodiments theprovided non-evanescent electromagnetic wave is a freely-propagatingelectromagnetic wave, e.g. a wave that is transmitted by and freelyradiates from the second surface region (including diverging propagatingwaves, converging propagating waves, and substantially planarpropagating waves). In other embodiments the provided non-evanescentwave is a confinedly-propagating electromagnetic wave, e.g. a wave thatis transmitted by the second surface region into a propagating guidedwave mode (as in a waveguide, transmission line, optical fiber, etc.)While the first and second surface regions 111 and 112 are depicted inFIG. 1 as exterior surfaces of the conversion structure 100, in otherembodiments the first surface region and/or the second surface regionmay be at least partially interior to the conversion structure (e.g.where the conversion structure includes one or more of a refractivecladding, an impedance-matching layer, input or output opticalcomponents, etc.). The use of a ray description, in FIG. 1 andelsewhere, is a heuristic convenience for purposes of visualillustration, and is not intended to connote any limitations orassumptions of geometrical optics; further, the elements depicted inFIG. 1 can have spatial dimensions that are variously less than, greaterthan, or comparable to a wavelength of interest. At the first surfaceregion 111, the evanescent electromagnetic wave 120 may be characterizedby a first transverse wavevector k_(T) ⁽¹⁾ (corresponding to a surfaceparallel direction of the first surface region indicated as the vectors141 in FIG. 1) that exceeds a first maximum transverse wavevectork_(max) ⁽¹⁾ for non-evanescent waves (cf. equation (2) and relatedpreceding text):

$\begin{matrix}{{k_{T}^{(1)} > k_{\max}^{(1)}} = {\frac{2\pi \; n_{1}f}{c} = \frac{2\pi \; f}{v_{1}}}} & (23)\end{matrix}$

where f is the frequency of the evanescent electromagnetic wave and v₁is a phase velocity (at the frequency f) for electromagnetic waves in afirst region outside the conversion structure 100 and adjacent to thefirst surface region 111 (the phase velocity may correspond to an indexof refraction n₁ for a refractive medium, possibly vacuum, in the firstregion, according to the relation v₁=c/n₁). At the second surface region112, the non-evanescent electromagnetic wave 130 may be characterized bya second transverse wavevector k_(T) ⁽²⁾ (corresponding to a surfaceparallel direction of the second surface region indicated as the vectors142 in FIG. 1) that does not exceed a second maximum transversewavevector k_(max) ⁽²⁾ for non-evanescent waves (cf. equation (2) andrelated preceding text):

$\begin{matrix}{{k_{T}^{(2)} < k_{\max}^{(2)}} = {\frac{2\pi \; n_{2}f}{c} = \frac{2\pi \; f}{v_{2}}}} & (24)\end{matrix}$

where f is the frequency of the non-evanescent electromagnetic wave andv₂ is a phase velocity (at the frequency f ) for electromagnetic wavesin a second region outside the conversion structure 100 and adjacent tothe second surface region 112 (the phase velocity may correspond to anindex of refraction n₂ for a refractive medium, possibly vacuum, in thesecond region, according to the relation v₂=c/n₂, where n₂ may be equalto or different than n₁).

In the illustrative embodiment of FIG. 1, the conversion structure 100has indefinite electromagnetic parameters, i.e. the conversion structureprovides an indefinite medium (i.e. an electromagnetic medium having anindefinite permittivity and/or an indefinite permeability, as discussedabove) that is responsive to the evanescent electromagnetic wave 120 toconvey a propagating electromagnetic wave from the first surface region111 to the second surface region 112. The indefinite medium defines anaxial direction (indicated by the vectors 150 at various positionswithin the indefinite medium), which, as previously discussed,corresponds to a first eigenvector of the indefinite permittivity matrixand/or the indefinite permeability matrix; and the indefinite mediumfurther defines a transverse direction (indicated by the vectors 151 atvarious positions within the indefinite medium) that is perpendicular tothe axial direction and corresponds to a second eigenvector of theindefinite permittivity matrix and/or the indefinite permeabilitymatrix. In the illustrative embodiment of FIG. 1, the axial direction150 is a non-constant axial direction that is a function of locationwithin the conversion structure 100, i.e. the axial direction may beregarded as a vector field (a vector-valued function of location).Moreover, the axial direction is generally directed from the firstsurface region 111 to the second surface region 112, i.e. axial fieldlines corresponding to the axial direction vector field extend from thefirst surface region to the second surface region. In FIG. 1, the dashedrays 125, indicating the propagating electromagnetic wave, alsocorrespond to axial field lines, because the illustrative embodimentdepicts a degenerate indefinite medium, i.e. an indefinite medium, asdescribed previously, that substantially conveys electromagnetic energyalong a propagation direction that corresponds to the axial direction ofthe indefinite medium. (This depiction is not intended to be limiting:in other embodiments, the indefinite medium is a “non-degenerate”indefinite medium that substantially conveys electromagnetic energyalong multiple propagation directions—e.g. along at least twopropagation directions, each of the at least two directions having asubstantially common angle with respect to the axial direction, or alonga plurality of propagation directions, the plurality of propagationdirections substantially composing a cone having a cone axis thatsubstantially coincides with the axial direction.)

Referring again to FIG. 1, the propagating electromagnetic field 125 maybe characterized by a transverse wavevector k_(T) that corresponds tothe transverse direction 151. In the present example, the axial fieldlines (corresponding to the vector field that describes the axialdirection 150) diverge geometrically as they proceed from the firstsurface region 111 to the second surface region 112, and this geometricdivergence may provide a substantially continuous variation of thetransverse wavevector k_(T), from a first transverse wavevector k_(T)⁽¹⁾ at the first surface region (as in equation (23), to match thetransverse wavevector of the evanescent electromagnetic wave 120) to asecond transverse wavevector k_(T) ⁽²⁾ at the second surface region (asin equation (24), to match the transverse wavevector of thenon-evanescent electromagnetic wave 130). Thus, the geometric divergenceof the axial field lines admits the conversion of an evanescentelectromagnetic wave 120 to a non-evanescent electromagnetic wave 130,by supporting a propagating electromagnetic wave 125 having asubstantially continuous variation of transverse wavevector from aninitial transverse wavevector that exceeds a maximum wavevector fornon-evanescent waves to a final transverse wavevector that does notexceed a maximum wavevector for non-evanescent waves.

With reference now to FIGS. 2 and 3, layered structures are depicted asexemplary implementations of the conversion structure 100 of FIG. 1. Inthe exemplary implementations of FIGS. 2 and 3, the conversion structure100 includes (as above) a first surface region 111 and a second surfaceregion 112 that are substantially planar and substantially parallel;intermediate the first and second surface regions, a layered structureprovides an effective continuous medium that corresponds to anindefinite medium. The layered structure includes layers of a firstmaterial 201 interleaved with layers of a second material 202, where thefirst and second materials have electromagnetic parameters (e.g.permittivities and/or permeabilities) that are oppositely-signed, asdescribed previously. In the exemplary implementations of FIGS. 2 and 3,the alternating Layers 201 and 202 are substantially nonplanar, having alayer normal direction that varies with position throughout the layeredstructure (i.e. from layer to layer and/or along the transverse extentof each layer), and this layer normal direction corresponds to the axialdirection (as depicted by the vectors 150 in FIG. 1) of the providedindefinite medium (equivalently, regarding the interfaces betweenalternating layers 201 and 202 as equipotential surfaces of a scalarfunction Φ, the gradient of Φ is locally parallel to the axial direction150 as per equation (22)). In the exemplary implementation of FIG. 3, afirst layer 301 of the layered structure substantially coincides withthe first surface region 111, and a last layer 302 of the layeredstructure substantially coincides with the second surface region 112,but this is not intended to be limiting (e.g. in the exemplaryimplementation of FIG. 2, neither the first surface region 111 nor thesecond surface region 112 substantially coincides with a layer of thelayered structure). The nonplanar alternating layers may havesubstantially uniform thickness throughout the transverse extents of thelayers, as in FIG. 2; or substantially non-uniform thicknessesthroughout the transverse extents of the layers, as in FIG. 3; or acombination thereof.

With reference now to FIG. 4, an illustrative embodiment is depictedthat includes a conversion structure 100 with indefinite electromagneticparameters, the conversion structure having a first surface region 111that is substantially planar and a second surface region 112 that issubstantially nonplanar. In the illustrative embodiment, thesubstantially nonplanar second surface region 112 is depicted as aconvex surface region (i.e. the configuration is a “plano-convex”configuration), but this is an exemplary configuration and is notintended to be limiting: other embodiments (not depicted) provide asubstantially nonplanar second surface region 112 that is concave (a“plano-concave” configuration), or that includes a first subregion thatis concave and a second subregion that is convex. This and otherdrawings, unless context dictates otherwise, can represent a planar viewof a three-dimensional embodiment, or a two-dimensional embodiment (e.g.in FIG. 4 where the conversion structure is placed inside a metallic ordielectric slab waveguide oriented normal to the page). The conversionstructure is responsive to an evanescent electromagnetic wave (depictedschematically as exponential tails 120) at the first surface region toconvey a propagating electromagnetic wave (depicted schematically asdashed rays 125) from the first surface region to the second surfaceregion, and to provide a non-evanescent electromagnetic wave (depictedschematically as the wavy rays 130) at the second surface region. Insome embodiments the provided non-evanescent electromagnetic wave is afreely-propagating electromagnetic wave, e.g. a wave that is transmittedby and freely radiates from the second electromagnetic surface(including diverging propagating waves, converging propagating waves,and substantially planar propagating waves). In other embodiments theprovided non-evanescent wave is a confinedly-propagating electromagneticwave, e.g. a wave that is transmitted by the second electromagneticsurface into a propagating guided wave mode (as in a waveguide,transmission line, optical fiber, etc.) While the first and secondsurface regions 111 and 112 are depicted in FIG. 4 as exterior surfacesof the conversion structure 100, in other embodiments the first surfaceregion and/or the second surface region may be at least partiallyinterior to the conversion structure (e.g. where the conversionstructure includes one or more of a refractive cladding, animpedance-matching layer, input or output optical components, etc.). Theuse of a ray description, in FIG. 4 and elsewhere, is a heuristicconvenience for purposes of visual illustration, and is not intended toconnote any limitations or assumptions of geometrical optics; further,the elements depicted in FIG. 4 can have spatial dimensions that arevariously less than, greater than, or comparable to a wavelength ofinterest. At the first surface region 111, the evanescentelectromagnetic wave 120 may be characterized by a first transversewavevector k_(T) ⁽¹⁾ (corresponding to a surface parallel direction ofthe first surface region indicated as the vectors 141 in FIG. 4) thatexceeds a first maximum transverse wavevector k_(max) ⁽¹⁾ defined as inequation (23). At the second surface region 112, the non-evanescentelectromagnetic wave 130 may be characterized by a second transversewavevector k_(T) ⁽²⁾ (corresponding to a surface parallel direction ofthe second surface region indicated as the vectors 142 in FIG. 4) thatdoes not exceed a second maximum transverse wavevector k_(max) ⁽²⁾defined as in equation (24).

In the illustrative embodiment of FIG. 4, the conversion structure 100has indefinite electromagnetic parameters, i.e. the conversion structureprovides an indefinite medium (i.e. an electromagnetic medium having anindefinite permittivity and/or an indefinite permeability, as discussedabove) that is responsive to the evanescent electromagnetic wave 120 toconvey a propagating electromagnetic wave from the first surface region111 to the second surface region 112. The indefinite medium defines anaxial direction (indicated by the vectors 150 at various positionswithin the indefinite medium), which, as previously discussed,corresponds to a first eigenvector of the indefinite permittivity matrixand/or the indefinite permeability matrix; and the indefinite mediumfurther defines a transverse direction (indicated by the vectors 151 atvarious positions within the indefinite medium) that is perpendicular tothe axial direction and corresponds to a second eigenvector of theindefinite permittivity matrix and/or the indefinite permeabilitymatrix. In the illustrative embodiment of FIG. 4, the axial direction150 is a non-constant axial direction that is a function of locationwithin the conversion structure 100, i.e. the axial direction may beregarded as a vector field (a vector-valued function of location).Moreover, the axial direction is generally directed from the firstsurface region 111 to the second surface region 112, i.e. axial fieldlines corresponding to the axial direction vector field extend from thefirst surface region to the second surface region. In FIG. 4, the dashedrays 125, indicating the propagating electromagnetic wave, alsocorrespond to axial field lines, because the illustrative embodimentdepicts a degenerate indefinite medium, i.e. an indefinite medium, asdescribed previously, that substantially conveys electromagnetic energyalong a propagation direction that corresponds to the axial direction ofthe indefinite medium. (This depiction is not intended to be limiting:in other embodiments, the indefinite medium is a “non-degenerate”indefinite medium that substantially conveys electromagnetic energyalong multiple propagation directions—e.g. along at least twopropagation directions, each of the at least two directions having asubstantially common angle with respect to the axial direction, or alonga plurality of propagation directions, the plurality of propagationdirections substantially composing a cone having a cone axis thatsubstantially coincides with the axial direction.)

Referring again to FIG. 4, the propagating electromagnetic field 125 maybe characterized by a transverse wavevector k_(T) that corresponds tothe transverse direction 151. In the present example, the axial fieldlines (corresponding to the vector field that describes the axialdirection 150) diverge geometrically as they proceed from the firstsurface region 111 to the second surface region 112, and this geometricdivergence may provide a substantially continuous variation of thetransverse wavevector k_(T), from a first transverse wavevector k_(T)⁽¹⁾ at the first surface region (as in equation (23), to match thetransverse wavevector of the evanescent electromagnetic wave 120) to asecond transverse wavevector k_(T) ⁽²⁾ at the second surface region(asin equation (24), to match the transverse wavevector of thenon-evanescent electromagnetic wave 130). Thus, the geometric divergenceof the axial field lines admits the conversion of an evanescentelectromagnetic wave 120 to a non-evanescent electromagnetic wave 130,by supporting a propagating electromagnetic wave 125 having asubstantially continuous variation of transverse wavevector from aninitial transverse wavevector that exceeds a maximum wavevector fornon-evanescent waves to a final transverse wavevector that does notexceed a maximum wavevector for non-evanescent waves.

With reference now to FIGS. 5 and 6, layered structures are depicted asexemplary implementations of the conversion structure 100 of FIG. 4. Inthe exemplary implementations of FIGS. 5 and 6, the conversion structure100 includes (as in FIG. 4) a first surface region 111 that issubstantially planar and a second surface region 112 that substantiallynonplanar; intermediate the first and second surface regions, a layeredstructure provides an effective continuous medium that corresponds to anindefinite medium. The layered structure includes layers of a firstmaterial 201 interleaved with layers of a second material 202, where thefirst and second materials have electromagnetic parameters (e.g.permittivities and/or permeabilities) that are oppositely-signed, asdescribed previously. In the exemplary implementations of FIGS. 5 and 6,the alternating layers 201 and 202 are substantially nonplanar, having alayer normal direction that varies with position throughout the layeredstructure (i.e. from layer to layer and/or along the transverse extentof each layer), and this layer normal direction corresponds to the axialdirection (as depicted by the vectors 150 in FIG. 4) of the providedindefinite medium (equivalently, regarding the interfaces betweenalternating layers 201 and 202 as equipotential surfaces of a scalarfunction Φ, the gradient of Φ is locally parallel to the axial direction150 as per equation (22)). In the exemplary implementation of FIG. 6, afirst layer 301 of the layered structure substantially coincides withthe first surface region 111, and a last layer 302 of the layeredstructure substantially coincides with the second surface region 112,but this is not intended to be limiting (e.g. in the exemplaryimplementation of FIG. 5, only the second surface region 112substantially coincides with a layer 302 of the layered structure). Thenonplanar alternating layers may have substantially uniform thicknessthroughout the transverse extents of the layers, as in FIG. 5; orsubstantially non-uniform thicknesses throughout the transverse extentsof the layers, as in FIG. 6; or a combination thereof.

With reference now to FIG. 7, an illustrative embodiment is depictedthat includes a conversion structure 100 with indefinite electromagneticparameters, the conversion structure having a first surface region 111that is substantially nonplanar and a second surface region 112 that issubstantially planar. In the illustrative embodiment, the substantiallynonplanar first surface region 111 is depicted as a concave surfaceregion (i.e. the configuration is a “concave-plano” configuration), butthis is an exemplary configuration and is not intended to be limiting:other embodiments (not depicted) provide a substantially nonplanar firstsurface region 111 that is convex (a “convex-plano” configuration), orthat includes a first subregion that is concave and a second subregionthat is convex. This and other drawings, unless context dictatesotherwise, can represent a planar view of a three-dimensionalembodiment, or a two-dimensional embodiment (e.g. in FIG. 7 where theconversion structure is placed inside a metallic or dielectric slabwaveguide oriented normal to the page). The conversion structure isresponsive to an evanescent electromagnetic wave (depicted schematicallyas exponential tails 120) at the first surface region to convey apropagating electromagnetic wave (depicted schematically as dashed rays125) from the first surface region to the second surface region, and toprovide a non-evanescent electromagnetic wave (depicted schematically asthe wavy rays 130) at the second surface region. In some embodiments theprovided non-evanescent electromagnetic wave is a freely-propagatingelectromagnetic wave, e.g. a wave that is transmitted by and freelyradiates from the second electromagnetic surface (including divergingpropagating waves, converging propagating waves, and substantiallyplanar propagating waves). In other embodiments the providednon-evanescent wave is a confinedly-propagating electromagnetic wave,e.g. a wave that is transmitted by the second electromagnetic surfaceinto a propagating guided wave mode (as in a waveguide, transmissionline, optical fiber, etc.) While the first and second surface regions111 and 112 are depicted in FIG. 7 as exterior surfaces of theconversion structure 100, in other embodiments the first surface regionand/or the second surface region may be at least partially interior tothe conversion structure (e.g. where the conversion structure includesone or more of a refractive cladding, an impedance-matching layer, inputor output optical components, etc.). The use of a ray description, inFIG. 7 and elsewhere, is a heuristic convenience for purposes of visualillustration, and is not intended to connote any limitations orassumptions of geometrical optics; further, the elements depicted inFIG. 7 can have spatial dimensions that are variously less than, greaterthan, or comparable to a wavelength of interest. At the first surfaceregion 111, the evanescent electromagnetic wave 120 may be characterizedby a first transverse wavevector k_(T) ⁽¹⁾ (corresponding to a surfaceparallel direction of the first surface region indicated as the vectors141 in FIG. 7) that exceeds a first maximum transverse wavevectork_(max) ⁽¹⁾ defined as in equation (23). At the second surface region112, the non-evanescent electromagnetic wave 130 may be characterized bya second transverse wavevector k_(T) ⁽²⁾ (corresponding to a surfaceparallel direction of the second surface region indicated as the vectors142 in FIG. 7) that does not exceed a second maximum transversewavevector k_(max) ⁽²⁾ defined as in equation (24).

In the illustrative embodiment of FIG. 7, the conversion structure 100has indefinite electromagnetic parameters, i.e. the conversion structureprovides an indefinite medium (i.e. an electromagnetic medium having anindefinite permittivity and/or an indefinite permeability, as discussedabove) that is responsive to the evanescent electromagnetic wave 120 toconvey a propagating electromagnetic wave from the first surface region111 to the second surface region 112. The indefinite medium defines anaxial direction (indicated by the vectors 150 at various positionswithin the indefinite medium), which, as previously discussed,corresponds to a first eigenvector of the indefinite permittivity matrixand/or the indefinite permeability matrix; and the indefinite mediumfurther defines a transverse direction (indicated by the vectors 151 atvarious positions within the indefinite medium) that is perpendicular tothe axial direction and corresponds to a second eigenvector of theindefinite permittivity matrix and/or the indefinite permeabilitymatrix. In the illustrative embodiment of FIG. 7, the axial direction150 is a non-constant axial direction that is a function of locationwithin the conversion structure 100, i.e. the axial direction may beregarded as a vector field (a vector-valued function of location).Moreover, the axial direction is generally directed from the firstsurface region 111 to the second surface region 112, i.e. axial fieldlines corresponding to the axial direction vector field extend from thefirst surface region to the second surface region. In FIG. 7, the dashedrays 125, indicating the propagating electromagnetic wave, alsocorrespond to axial field lines, because the illustrative embodimentdepicts a degenerate indefinite medium, i.e. an indefinite medium, asdescribed previously, that substantially conveys electromagnetic energyalong a propagation direction that corresponds to the axial direction ofthe indefinite medium. (This depiction is not intended to be limiting:in other embodiments, the indefinite medium is a “non-degenerate”indefinite medium that substantially conveys electromagnetic energyalong multiple propagation directions—e.g. along at least twopropagation directions, each of the at least two directions having asubstantially common angle with respect to the axial direction, or alonga plurality of propagation directions, the plurality of propagationdirections substantially composing a cone having a cone axis thatsubstantially coincides with the axial direction.)

Referring again to FIG. 7, the propagating electromagnetic field 125 maybe characterized by a transverse wavevector k_(T) that corresponds tothe transverse direction 151. In the present example, the axial fieldlines (corresponding to the vector field that describes the axialdirection 150) diverge geometrically as they proceed from the firstsurface region 111 to the second surface region 112, and this geometricdivergence may provide a substantially continuous variation of thetransverse wavevector k_(T), from a first transverse wavevector k_(T)⁽¹⁾ at the first surface region (as in equation (23), to match thetransverse wavevector of the evanescent electromagnetic wave 120) to asecond transverse wavevector k_(T) ⁽²⁾ at the second surface region (asin equation (24), to match the transverse wavevector of thenon-evanescent electromagnetic wave 130). Thus, the geometric divergenceof the axial field lines admits the conversion of an evanescentelectromagnetic wave 120 to a non-evanescent electromagnetic wave 130,by supporting a propagating electromagnetic wave 125 having asubstantially continuous variation of transverse wavevector from aninitial transverse wavevector that exceeds a maximum wavevector fornon-evanescent waves to a final transverse wavevector that does notexceed a maximum wavevector for non-evanescent waves.

With reference now to FIGS. 8 and 9, layered structures are depicted asexemplary implementations of the conversion structure 100 of FIG. 7. Inthe exemplary implementations of FIGS. 8 and 8, the conversion structure100 includes (as in FIG. 7) a first surface region 111 that issubstantially nonplanar and a second surface region 112 thatsubstantially planar; intermediate the first and second surface regions,a layered structure provides an effective continuous medium thatcorresponds to an indefinite medium. The layered structure includeslayers of a first material 201 interleaved with layers of a secondmaterial 202, where the first and second materials have electromagneticparameters (e.g. permittivities and/or permeabilities) that areoppositely-signed, as described previously. In the exemplaryimplementations of FIGS. 8 and 9, the alternating layers 201 and 202 aresubstantially nonplanar, having a layer normal direction that varieswith position throughout the layered structure (i.e. from layer to layerand/or along the transverse extent of each layer), and this layer normaldirection corresponds to the axial direction (as depicted by the vectors150 in FIG. 7) of the provided indefinite medium (equivalently,regarding the interfaces between alternating layers 201 and 202 asequipotential surfaces of a scalar function Φ, the gradient of Φ islocally parallel to the axial direction 150 as per equation (22)). Inthe exemplary implementation of FIG. 9, a first layer 301 of the layeredstructure substantially coincides with the first surface region 111, anda last layer 302 of the layered structure substantially coincides withthe second surface region 112, but this is not intended to be limiting(e.g. in the exemplary implementation of FIG. 8, only the first surfaceregion 112 substantially coincides with a layer 301 of the layeredstructure). The nonplanar alternating layers may have substantiallyuniform thickness throughout the transverse extents of the layers, as inFIG. 8; or substantially non-uniform thicknesses throughout thetransverse extents of the layers, as in FIG. 9; or a combinationthereof.

With reference now to FIG. 10, various illustrative embodiments aredepicted that include a conversion structure 100 with indefiniteelectromagnetic parameters, the conversion structure having a firstsurface region 111 and a second surface region 112, the first surfaceregion and the second surface region being substantially nonplanar andsubstantially non-concentric. In general, a first surface region and asecond surface region are non-concentric if: the first surface regionhas a non-constant curvature, and/or the second surface region has anon-constant curvature, and/or a center of an osculating circle of thefirst surface region is different than a center of an osculating circleof the second surface region (where the osculating circles arecoplanar). Embodiment 1001 depicts a conversion structure 100 having afirst surface region 111 and a second surface region 112 that are bothconvex (a “bi-convex” configuration). Embodiment 1002 depicts aconversion structure 100 having a first surface region 111 and a secondsurface region 112 that are both concave (a “bi-concave” configuration).Embodiment 1003 depicts a conversion structure 100 having a firstsurface region 111 that is concave and a second surface region 112 thatis convex, where a center of curvature of the first surface region is tothe right of—i.e. nearer to the conversion structure than—a center ofcurvature of the second surface region, i.e. a concave-convex “negativemeniscus” configuration (another exemplary embodiment-not shown-providesa concave-convex “positive meniscus” configuration where the center ofcurvature of the first surface region is to the left—i.e. farther fromthe conversion structure than—the center of curvature of the secondsurface region). Embodiment 1004 depicts a conversion structure 100having a first surface region 111 that is convex and a second surfaceregion 112 that is concave, where a center of curvature of the firstsurface region is to the left of—i.e. nearer to the conversion structurethan—a center of curvature of the second surface region, i.e. aconvex-concave “positive meniscus” configuration (another exemplaryembodiment-not shown-provides a convex-concave “negative meniscus”configuration where the center of curvature of the first surface regionis to the right of—i.e. farther from the conversion structure than—thecenter of curvature of the second surface region). Embodiment 1005depicts a conversion structure having a first surface region 111 that ispartially convex and partially concave and a second surface region 112that is partially convex and partially concave (in other embodiments,not shown, the first surface region 111 is convex only or concave onlyand the second surface region 112 is partially convex and partiallyconcave, or the first surface region 111 is partially convex andpartially concave and the second surface region 112 is convex only orconcave only). As elsewhere, the depictions in FIG. 10 can representplanar views of three-dimensional embodiments, or a two-dimensionalembodiment (e.g. where the conversion structure 100 is placed inside ametallic or dielectric slab waveguide oriented normal to the page). Ineach embodiment of FIG. 10, the conversion structure 100 is responsiveto an evanescent electromagnetic wave (depicted schematically asexponential tails 120) at the first surface region 111 to convey apropagating electromagnetic wave (depicted schematically as dashed rays125) from the first surface region to the second surface region, and toprovide a non-evanescent electromagnetic wave (depicted schematically asthe wavy rays 130) at the second surface region 112. In someapplications the provided non-evanescent electromagnetic wave is afreely-propagating electromagnetic wave, e.g. a wave that is transmittedby and freely radiates from the second electromagnetic surface(including diverging propagating waves, converging propagating waves,and substantially planar propagating waves). In other applications theprovided non-evanescent wave is a confinedly-propagating electromagneticwave, e.g. a wave that is transmitted by the second electromagneticsurface into a propagating guided wave mode (as in a waveguide,transmission line, optical fiber, etc.) While the first and secondsurface regions 111 and 112 are depicted in FIG. 10 as exterior surfacesof the conversion structure 100, in other embodiments the first surfaceregion and/or the second surface region may be at least partiallyinterior to the conversion structure (e.g. where the conversionstructure includes one or more of a refractive cladding, animpedance-matching layer, input or output optical components, etc.). Theuse of a ray description, in FIG. 10 and elsewhere, is a heuristicconvenience for purposes of visual illustration, and is not intended toconnote any limitations or assumptions of geometrical optics; further,the elements depicted in FIG. 10 can have spatial dimensions that arevariously less than, greater than, or comparable to a wavelength ofinterest. At the first surface region 111, the evanescentelectromagnetic wave 120 may be characterized by a first transversewavevector k_(T) ⁽¹⁾ (corresponding to a surface parallel direction ofthe first surface region—for simplicity this surface parallel directionof the first surface region is not depicted in the embodiments of FIG.10, but should be apparent from the analogous elements 141 depicted inFIGS. 1, 4, and 7) that exceeds a first maximum transverse wavevectork_(max) ⁽¹⁾ defined as in equation (23). At the second surface region112, the non-evanescent electromagnetic wave 130 may be characterized bya second transverse wavevector k_(T) ⁽²⁾ (corresponding to a surfaceparallel direction of the second surface region-again for simplicitythis surface parallel direction of the second surface region is notdepicted in the embodiments of FIG. 10, but should be apparent from theanalogous elements 142 depicted in FIGS. 1, 4, and 7) that does notexceed a second maximum transverse wavevector k_(max) ⁽²⁾ defined as inequation (24).

In the illustrative embodiments of FIG. 10, the conversion structure 100has indefinite electromagnetic parameters, i.e. the conversion structureprovides an indefinite medium (i.e. an electromagnetic medium having anindefinite permittivity and/or an indefinite permeability, as discussedabove) that is responsive to the evanescent electromagnetic wave 120 toconvey a propagating electromagnetic wave from the first surface region111 to the second surface region 112. The indefinite medium defines anaxial direction (again for simplicity this axial direction is notdepicted in the embodiments of FIG. 10, but should be apparent from theanalogous elements 150 depicted in FIGS. 1, 4, and 7), which, aspreviously discussed, corresponds to a first eigenvector of theindefinite permittivity matrix and/or the indefinite permeabilitymatrix; and the indefinite medium further defines a transverse direction(again for simplicity this transverse direction is not depicted in theembodiments of FIG. 10, but should be apparent from the analogouselements 151 depicted in FIGS. 1, 4, and 7) that is perpendicular to theaxial direction and corresponds to a second eigenvector of theindefinite permittivity matrix and/or the indefinite permeabilitymatrix. In the illustrative embodiments of FIG. 10, the axial direction150 is a non-constant axial direction that is a function of locationwithin the conversion structure 100, i.e. the axial direction may beregarded as a vector field (a vector-valued function of location).Moreover, the axial direction is generally directed from the firstsurface region 111 to the second surface region 112, i.e. axial fieldlines corresponding to the axial direction vector field extend from thefirst surface region to the second surface region. In FIG. 1, the dashedrays 125, indicating the propagating electromagnetic wave, alsocorrespond to axial field lines, because the illustrative embodimentdepicts a degenerate indefinite medium, i.e. an indefinite medium, asdescribed previously, that substantially conveys electromagnetic energyalong a propagation direction that corresponds to the axial direction ofthe indefinite medium. (This depiction is not intended to be limiting:in other embodiments, the indefinite medium is a “non-degenerate”indefinite medium that substantially conveys electromagnetic energyalong multiple propagation directions—e.g. along at least twopropagation directions, each of the at least two directions having asubstantially common angle with respect to the axial direction, or alonga plurality of propagation directions, the plurality of propagationdirections substantially composing a cone having a cone axis thatsubstantially coincides with the axial direction.)

Referring again to FIG. 10, the propagating electromagnetic field 125may be characterized by a transverse wavevector k_(T) that correspondsto the transverse direction (again for simplicity this transversedirection is not depicted in the embodiments of FIG. 10, but should beapparent from the analogous elements 151 depicted in FIGS. 1, 4, and 7).In the present example, the axial field lines (corresponding to thevector field that describes the axial direction) diverge geometricallyas they proceed from the first surface region 111 to the second surfaceregion 112, and this geometric divergence may provide a substantiallycontinuous variation of the transverse wavevector k_(T), from a firsttransverse wavevector k_(T) ⁽¹⁾ at the first surface region (as inequation (23), to match the transverse wavevector of the evanescentelectromagnetic wave 120) to a second transverse wavevector k_(t) ⁽²⁾ atthe second surface region(as in equation (24), to match the transversewavevector of the non-evanescent electromagnetic wave 130). Thus, thegeometric divergence of the axial field lines admits the conversion ofan evanescent electromagnetic wave 120 to a non-evanescentelectromagnetic wave 130, by supporting a propagating electromagneticwave 125 having a substantially continuous variation of transversewavevector from an initial transverse wavevector that exceeds a maximumwavevector for non-evanescent waves to a final transverse wavevectorthat does not exceed a maximum wavevector for non-evanescent waves.

With reference now to FIG. 11, a layered structure is depicted as anexemplary implementation of a conversion structure 100 as in FIG. 10. Inthe exemplary implementation of FIG. 11, the conversion structure 100includes a first surface region 111 and a second surface region 112 thatare substantially nonplanar and substantially non-concentric;intermediate the first and second surface regions, a layered structureprovides an effective continuous medium that corresponds to anindefinite medium. The layered structure includes layers of a firstmaterial 201 interleaved with layers of a second material 202, where thefirst and second materials have electromagnetic parameters (e.g.permittivities and/or permeabilities) that are oppositely-signed, asdescribed previously. In the exemplary implementations of FIG. 11, thealternating layers 201 and 202 are substantially nonplanar, having alayer normal direction that varies with position throughout the layeredstructure (i.e. from layer to layer and/or along the transverse extentof each layer), and this layer normal direction corresponds to the axialdirection of the provided indefinite medium (equivalently, regarding theinterfaces between alternating layers 201 and 202 as equipotentialsurfaces of a scalar function Φ, the gradient of Φ is locally parallelto the axial direction as per equation (22)). In the exemplaryimplementation of FIG. 3, a first layer 301 of the layered structuresubstantially coincides with the first surface region 111, and a lastlayer 302 of the layered structure substantially coincides with thesecond surface region 112, but this is not intended to be limiting (inother embodiments, not depicted, the first surface region 111 does notcoincide with a layer of the layered structure, and/or the secondsurface region 112 does not coincide with a layer of the layeredstructure). The nonplanar alternating layers may have substantiallyuniform thickness throughout the transverse extents of the layers; orsubstantially non-uniform thicknesses throughout the transverse extentsof the layers; or a combination thereof, as in FIG. 11.

In some embodiments a conversion structure, such as those depicted inFIGS. 1-11, includes an indefinite transformation medium, i.e. atransformation medium that has indefinite electromagnetic parameters.For example, the geometric divergence of the axial field lines, alongthe dashed rays 125 in FIGS. 1, 4, 7, and 10, may accord with acoordinate transformation, e.g. from an untransformed coordinate spacein which the axial field lines do not have a geometric divergence.Recalling the previous exemplary planar slab of indefinite medium (i.e.as described by equations (3), (5), (9), and/or (11) and the textaccompanying these equations), the planar slab has an axial directionthat corresponds to the z-axis; thus, axial field lines for the planarslab are straight lines parallel to the z-axis and perpendicular to thefaces of the slab (for purposes of illustration, suppose that the facesof the slab—its first and second surface regions—are located at z=0 andz=d, respectively). To obtain an indefinite transformation medium,suppose that this planar slab is regarded as the original untransformedmedium in equations (15) and (16), and consider an exemplary coordinatetransformation that maps a portion of the surface z=0 to the firstsurface region 111 of the conversion structure 100, and a portion of thesurface z=d to the second surface region 112 of the conversion structure100, with portions of the intermediate surfaces of constant z (i.e. for0<z<d) mapped to successive surfaces S_(z) in the transformed coordinatespace, so that the family of successive surfaces {S_(z)|0<z<d} spans aregion intermediate the first surface region and the second surfaceregion in the transformed coordinate space. Further, suppose that theexemplary coordinate transformation provides an increasing magnificationof the successive constant-z surfaces according to a magnificationfactor m(z) so that, for example, two lines that are parallel to thez-axis in the untransformed coordinate space shall diverge in thetransformed coordinate space, with a geodesic distance between the twolines being proportional to m(z) on the surface S_(z). For theconversion structure 100 of FIG. 1, having first and second surfaceregions 111 and 112 that are substantially parallel and substantiallyplanar, an exemplary coordinate transformation maps planes of constant zto planes of constant z′ according to the equations

x′=m(z)x

y′=m(z)y

z′=z   (25)

where m(z) is a magnification factor that increases with z (e.g. fromm=1 at the first surface region to m=M>1 at the second surface region).In this example, the first and second surface regions of the indefinitetransformation medium, at z′=0 and z′=d, respectively, correspond to thefirst and second surface regions 111 and 112 of the conversion structure100 in FIG. 1. Constitutive parameters of the indefinite transformationmedium (obtained from equations (15) and (16) with the Jacobian matrix(17) corresponding to coordinate transformation (25)) provide anindefinite medium with axial field lines that diverge geometrically asthey proceed from the first surface region to the second surface region,in accordance with the magnifying coordinate transformation (25). Then,for a propagating electromagnetic wave 125 characterized by a transversewavevector k_(T), the transverse wavevector varies in inverse proportionto the magnification factor m(z′) as the propagating electromagneticwave advances from the first surface region 111 to the second surfaceregion 112, implying the connection

$\begin{matrix}{\frac{k_{T}^{(1)}}{k_{T}^{(2)}} = {\frac{m\left( {z^{\prime} = d} \right)}{m\left( {z^{\prime} = 0} \right)} = M}} & (26)\end{matrix}$

between the first transverse wavevector k_(T) ⁽¹⁾ for the evanescentelectromagnetic wave 120 (at the first surface region) and the secondtransverse wavevector k_(T) ⁽²⁾ for the non-evanescent electromagneticwave 130 (at the second surface region). Therefore the conversionstructure 100 will convert an evanescent electromagnetic wave 120 to anon-evanescent electromagnetic wave 130 for a range of transversewavevectors k_(T) ⁽¹⁾ ∈ (k_(max) ⁽¹⁾,Mk_(max) ⁽²⁾) (cf. equations (23)and (24)); or, reciprocally, the conversion structure 100 will convert anon-evanescent electromagnetic wave 130 to an evanescent electromagneticwave 120 for a range of transverse wavevectors k_(T) ⁽²⁾ ∈ (M⁻¹k_(max)⁽¹⁾,k_(max) ⁽²⁾).

In some embodiments, the planar slab of untransformed indefinite mediumis a degenerate indefinite medium, i.e. providing degenerate propagationfor TM-polarized waves (with |ε_(x)| and/or |ε_(y)| substantially lessthan |ε_(z)|), TE-polarized waves (with |μ_(x)| and/or |μ_(y)|substantially less than |μ_(z)|), or both. For example, the planar slabmay have a permittivity matrix

$\begin{matrix}{ɛ = {\begin{pmatrix}ɛ_{x} & 0 & 0 \\0 & ɛ_{y} & 0 \\0 & 0 & ɛ_{z}\end{pmatrix} \approx \begin{pmatrix}0 & 0 & 0 \\0 & 0 & 0 \\0 & 0 & ɛ_{z}\end{pmatrix}}} & (27)\end{matrix}$

(where the symbol “≈” indicates that the transverse components areapproximated as zero). In the transformed coordinate space, the newpermittivity tensor is

ε ^(i′j′) _(≈|det(Λ)|) ⁻¹Λ_(z) ^(i′)Λ_(z) ^(j′)ε_(z)   (28)

which may be diagonalized in the new coordinate space as

$\begin{matrix}{\overset{\sim}{ɛ} \approx {\frac{ɛ_{z}}{{\det (\Lambda)}}{\begin{pmatrix}0 & 0 & 0 \\0 & 0 & 0 \\0 & 0 & {\sum\limits_{i^{\prime}}{\Lambda_{z}^{i^{\prime}}\Lambda_{z}^{i^{\prime}}}}\end{pmatrix}.}}} & (29)\end{matrix}$

The transformation medium is a new degenerate indefinite medium, with anew spatially-varying axial direction given by

$\begin{matrix}{u_{A} \propto \left( {\frac{\partial x^{\prime}}{\partial z},\frac{\partial y^{\prime}}{\partial z},\frac{\partial z^{\prime}}{\partial z}} \right) \propto \left( {x^{\prime},y^{\prime},\frac{m\left( z^{\prime} \right)}{m^{\prime}\left( z^{\prime} \right)}} \right)} & (30)\end{matrix}$

(in the coordinate basis (x′,y′,z′)), where the latter proportionalityis obtained by substituting equation (25). In some embodiments thistransformation medium may be implemented as a nonplanar layeredstructure (cf. the preceding discussion of layered structures), byrelating the vector field (30) to a scalar potential Φ according toequation (22) whereby the interfaces of adjacent layers in the nonplanarlayered structure correspond to equipotential surfaces of the functionΦ. In a first example, the magnification factor may increase linearlywith z, e.g.

$\begin{matrix}{{{m(z)} = {1 + {\left( {M - 1} \right)\frac{z}{d}}}};} & (31)\end{matrix}$

the resultant axial vector field (30) corresponds to a scalar potentialΦ having equipotential surfaces that are concentric spheres (orcylinders, in a two-dimensional embodiment) centered at z′=−d/(M−1). Thelayered structure of FIG. 2 resembles a configuration of this sort;moreover the layered structures of FIGS. 5 and 8 resemble theconfiguration of FIG. 2, absent selected layers so as to have either anonplanar first surface region or a nonplanar second surface region ofthe conversion structure, but providing similar indefinite mediumproperties within the interior of the conversion structure. In a secondexample, the magnification factor may increase nonlinearly with z, e.g.

$\begin{matrix}{{m(z)} = {1 + {\frac{M - 1}{2}\left( {1 - {\cos \frac{\pi \; z}{d}}} \right)}}} & (32)\end{matrix}$

(the functional dependence being selected to have m′(0)=m′(d)=0); theresultant axial vector field (30) corresponds to a scalar potential Φhaving successive equipotential surfaces that evolve from a planarsurface at z′=0 through a series of curved surfaces to another planarsurface at z′=d. The layered structure of FIG. 3 resembles aconfiguration of this sort; moreover the layered structures of FIGS. 6and 9 resemble the configuration of FIG. 3, absent selected layers so asto have either a nonplanar first surface region or a nonplanar secondsurface region of the conversion structure, but providing similarindefinite medium properties within the interior of the conversionstructure.

The exemplary conversion structures 100 in FIGS. 1, 4, 7, and 10 providean indefinite medium that is depicted as responding to an evanescentelectromagnetic wave to provide a non-evanescent electromagnetic wave.In some embodiments (e.g. where the indefinite medium is a reciprocalelectromagnetic medium) the indefinite medium alternately oradditionally has a reciprocal response, i.e. the indefinite mediumresponds to a non-evanescent electromagnetic wave to provide anevanescent electromagnetic wave. In a reciprocal response of aconversion structure 100 as in FIGS. 1, 4, 7, and 10, the non-evanescentelectromagnetic wave 130 and the propagating electromagnetic wave 125may be regarded as having spatially-reversed propagation directions(i.e. propagating from right to left in the figures) and the evanescentelectromagnetic wave 120 may be regarded as having a spatially-reversedexponential decay (i.e. having an exponential decay from right to left,rather than from left to right as depicted). Thus, in a reciprocalscenario, the conversion structure 100 is responsive to a(leftwards-propagating) non-evanescent electromagnetic wave 130 at thesecond surface region 112 to convey a propagating electromagnetic wave125 from the second surface region to the first surface region 111 andto provide an (leftwards-decaying) evanescent electromagnetic wave 120at the first surface region. As before, the evanescent electromagneticwave 120 may be characterized by a first transverse wavevector k_(T) ⁽¹⁾as in equation (23) and the non-evanescent electromagnetic wave 130 maybe characterized by a second transverse wavevector k_(T) ⁽²⁾ as inequation (24). In the reciprocal scenario, when the indefinite medium isa degenerate indefinite medium, the propagating electromagnetic wave 125may propagate along a propagation direction that corresponds to adirection antiparallel to the axial direction of the indefinite medium;when the indefinite medium is a non-degenerate indefinite medium, thepropagating electromagnetic wave 125 may propagate along multiplepropagation directions—e.g. along at least two propagation directions,each of the at least two directions having a substantially common anglewith respect to a direction antiparallel to the axial direction, oralong a plurality of propagation directions, the plurality ofpropagation directions substantially composing a cone having a cone axisthat substantially coincides with a direction antiparallel to the axialdirection. In the depictions of FIGS. 1, 4, 7, and 10, the axial fieldlines diverge geometrically as they proceed from the first surfaceregion 111 to the second surface region 112; equivalently, the axialfield lines converge geometrically from the second surface 112 to thefirst surface 111. In the reciprocal scenario, this geometricconvergence may provide—for a propagating electromagnetic wave 125characterized by a transverse wavevector k_(T)—a substantiallycontinuous variation of the transverse wavevector k_(T), from k_(T) ⁽²⁾at the second surface region (as in equation (24), to match thetransverse wavevector of the non-evanescent electromagnetic wave 130) tok_(T) ⁽¹⁾ at the first surface region (as in equation (23), to match thetransverse wavevector of the evanescent electromagnetic wave 120). Thus,the geometric convergence of the axial field lines admits the conversionof a non-evanescent electromagnetic wave 130 to an evanescentelectromagnetic wave 120, by supporting a propagating electromagneticwave having a substantially continuous variation of transversewavevector from an initial transverse wavevector that does not exceed amaximum wavevector for non-evanescent waves to a final transversewavevector that exceeds a maximum wavevector for non-evanescent waves.

Some embodiments are responsive to an evanescent electromagnetic wave toprovide a non-evanescent electromagnetic wave (and/or vice versa, in areciprocal scenario) at a selected frequency/frequency band and/or aselected polarization. The selected frequency or frequency band may beselected from a range that includes radio frequencies, microwavefrequencies, millimeter- or submillimeter-wave frequencies, THz-wavefrequencies, optical frequencies (e.g. variously corresponding to softx-rays, extreme ultraviolet, ultraviolet, visible, near-infrared,infrared, or far infrared light), etc. The selected polarization may bea TE polarization, a TM polarization, a circular polarization, etc.(other embodiments are responsive to an evanescent electromagnetic waveto provide a non-evanescent electromagnetic wave—and/or vice versa, in areciprocal scenario—for any polarization, e.g. for unpolarizedelectromagnetic energy).

Some embodiments are responsive to an evanescent electromagnetic wave toa provide a non-evanescent electromagnetic wave (and/or vice versa, in areciprocal scenario) at a first frequency, and further responsive to anevanescent electromagnetic wave to a provide a non-evanescentelectromagnetic wave (and/or vice versa, in a reciprocal scenario) at asecond frequency different than the first frequency. For embodimentsthat recite first and second frequencies, the first and secondfrequencies may be selected from the frequency categories in thepreceding paragraph. Moreover, for these embodiments, the recitation offirst and second frequencies may generally be replaced by a recitationof first and second frequency bands, again selected from the abovefrequency categories. These embodiments responsive at first and secondfrequencies may include a indefinite medium having adjustableelectromagnetic properties. For example, the indefinite medium may haveelectromagnetic properties that are adjustable (e.g. in response to anexternal input or control signal) between first electromagneticproperties and second electromagnetic properties, the firstelectromagnetic properties providing an indefinite medium responsive toan evanescent electromagnetic wave to provide a non-evanescentelectromagnetic wave (and/or vice versa) at the first frequency, and thesecond electromagnetic properties providing an indefinite mediumresponsive to an evanescent electromagnetic wave to provide anon-evanescent electromagnetic wave (and/or vice versa) at the secondfrequency. An indefinite medium with an adjustable electromagneticresponse may be implemented with variable metamaterials, e.g. asdescribed in R. A. Hyde et al, supra. Other embodiments responsive atfirst and second frequencies may include an indefinite medium having afrequency-dependent response to electromagnetic radiation, correspondingto frequency-dependent constitutive parameters. For example, thefrequency-dependent response at a first frequency may be a response toan evanescent electromagnetic wave to provide a non-evanescentelectromagnetic wave (and/or vice versa) at the first frequency, and thefrequency-dependent response at a second frequency may be a response toan evanescent electromagnetic wave to provide a non-evanescentelectromagnetic wave (and/or vice versa) at the second frequency. Anindefinite medium having a frequency-dependent response toelectromagnetic radiation can be implemented withartificially-structured materials such as metamaterials; for example, afirst set of metamaterial elements having a response at the firstfrequency may be interleaved with a second set of metamaterial elementshaving a response at the second frequency.

An illustrative embodiments is depicted as a process flow diagram inFIG. 12. Flow 1200 includes operation 1210—coupling to an evanescentelectromagnetic wave at an input region. For example, a conversionstructure, such as that depicted as element 100 in FIGS. 4 and 7,couples to an evanescent electromagnetic wave 120 at a first surfaceregion 111 of the conversion structure. Flow 1200 includes operation1220—responsive to the coupling, propagating electromagnetic energy inan electromagnetic field from a first surface within the input region toa second surface within an output region, where one of the first andsecond surfaces is a substantially planar surface and the other of thefirst and second surfaces is a substantially nonplanar surface. Forexample, the conversion structure 100 in FIGS. 4 and 7 conveys apropagating electromagnetic wave 125 from the first surface region 111to the second surface region 112. The first surface may be an exteriorsurface of the input region, or at least partially within an interiorportion of the input region (e.g. corresponding to a conversionstructure having a first surface region 111 that is at least partiallyinterior to the conversion structure). The second surface may be anexterior surface of the output region, or at least partially within aninterior portion of the output region (e.g. corresponding to aconversion structure having a second surface region 112 that is at leastpartially interior to the conversion structure).

Operation 1220 includes sub-operation 1221—inducing a first polarizationin a first direction, the first polarization positively corresponding toa first component of the electromagnetic field in the firstdirection-and sub-operation 1222—inducing a second polarization in asecond direction perpendicular to the first direction, the secondpolarization negatively corresponding to a second component of theelectromagnetic field in the second direction. For example, a conversionstructure 100 as in FIGS. 4 and 7 may provide an indefinite mediumhaving an indefinite permittivity, and the indefinite permittivity maycorrespond to an electric susceptibility that is positive in a firstdirection and negative in a second direction (where the first and seconddirections may correspond to the axial and transverse directions 150 and151 in FIGS. 4 and 7, or vice versa). Then the propagatingelectromagnetic wave 125 may induce an electric polarization in thefirst direction that positively corresponds (in accordance with thepositive electric susceptibility) to an electric field component of thepropagating electromagnetic wave in the first direction, and furtherinduce an electric polarization in the second direction that negativelycorresponds (in accordance with the negative electric susceptibility) toan electric field component of the propagating electromagnetic wave inthe second direction. In an alternate example, a conversion structure100 as in FIGS. 4 and 7 may provide an indefinite medium having anindefinite permeability, and the indefinite permeability may correspondto a magnetic susceptibility that is positive in a first direction andnegative in a second direction (where the first and second directionsmay correspond to the axial and transverse directions 150 and 151 inFIGS. 4 and 7, or vice versa). Then the propagating electromagnetic wave125 may induce a magnetic polarization in the first direction thatpositively corresponds (in accordance with the positive magneticsusceptibility) to a magnetic field component of the propagatingelectromagnetic wave in the first direction, and further induce amagnetic polarization in the second direction that negativelycorresponds (in accordance with the negative magnetic susceptibility) toa magnetic field component of the propagating electromagnetic wave inthe second direction. Operation 1220 optionally further includessub-operation 1223—inducing a third polarization in a third direction,the third polarization positively corresponding to a third component ofthe electromagnetic field in the third direction-and sub-operation1224—inducing a fourth polarization in a fourth direction perpendicularto the third direction, the fourth polarization negatively correspondingto a fourth component of the electromagnetic field in the fourthdirection. For example, a conversion structure 100 as in FIGS. 4 and 7may provide an indefinite medium having both an indefinite permittivityand an indefinite permeability, the indefinite permittivitycorresponding to a electric susceptibility that is positive in a firstdirection and negative in a second direction (where the first and seconddirections may correspond to the axial and transverse directions 150 and151 in FIGS. 4 and 7, or vice versa) and the indefinite permeabilitycorresponding to a magnetic susceptibility that is positive in a thirddirection and negative in a fourth direction (where the third and fourthdirections may correspond to the axial and transverse directions 150 and151 of FIGS. 4 and 7, or vice versa). Then the propagatingelectromagnetic wave 125 may induce: (1) an electric polarization in thefirst direction that positively corresponds (in accordance with thepositive electric susceptibility) to an electric field component of thepropagating electromagnetic wave in the first direction, (2) an electricpolarization in the second direction that negatively corresponds (inaccordance with the negative electric susceptibility) to an electricfield component of the propagating electromagnetic wave in the seconddirection, (3) a magnetic polarization in the third direction thatpositively corresponds (in accordance with the positive magneticsusceptibility) to a magnetic field component of the propagatingelectromagnetic wave in the second direction, and (4) a magneticpolarization in the fourth direction that negatively corresponds (inaccordance with the negative magnetic susceptibility) to a magneticfield component of the propagating electromagnetic wave in the fourthdirection. Flow 1200 includes operation 1230—providing the propagatedelectromagnetic energy as a non-evanescent electromagnetic wave at theoutput region. For example, the conversion structure 100 of FIGS. 4 and7 provides a non-evanescent electromagnetic wave 130 at the secondsurface region 112; the non-evanescent electromagnetic wave may be afreely-propagating electromagnetic wave, e.g. a wave that is emitted byand freely radiates from the second electromagnetic surface, or aconfinedly-propagating electromagnetic wave, e.g. a wave that istransmitted by the second surface region into a propagating guided wavemode (as in a waveguide, transmission line, optical fiber, etc.).

An illustrative embodiments is depicted as a process flow diagram inFIG. 13. Flow 1300 includes operation 1310—receiving a non-evanescentelectromagnetic wave at an input region. For example, a conversionstructure, such as that depicted as element 100 in FIGS. 4 and 7,receives (in a reciprocal scenario to that of FIG. 1, as describedpreviously) a non-evanescent electromagnetic wave 130 at the secondsurface region 112. Flow 1300 includes operation 1320—responsive to thereceiving, propagating electromagnetic energy in an electromagneticfield from a first surface within the input region to a second surfacewithin an output region, where one of the first and second surfaces is asubstantially planar surface and the other of the first and secondsurfaces is a substantially nonplanar surface. For example, theconversion structure 100 in FIGS. 4 and 7 conveys (in a reciprocalscenario to that of FIGS. 4 and 7, as described previously) apropagating electromagnetic wave 125 from the second surface region 112to the first surface region 111. The first surface may be an exteriorsurface of the input region, or at least partially within an interiorportion of the input region (e.g. corresponding to a conversionstructure having a second surface region 112 that is at least partiallyinterior to the conversion structure). The second surface may be anexterior surface of the output region, or at least partially within aninterior portion of the output region (e.g. corresponding to aconversion structure having an first surface region 111 that is at leastpartially interior to the conversion structure). Operation 1320 includessub-operation 1321—inducing a first polarization in a first direction,the first polarization positively corresponding to a first component ofthe electromagnetic field in the first direction-and sub-operation1322—inducing a second polarization in a second direction perpendicularto the first direction, the second polarization negatively correspondingto a second component of the electromagnetic field in the seconddirection. For example, a conversion structure 100 as in FIGS. 4 and 7may provide an indefinite medium having an indefinite permittivity, andthe indefinite permittivity may correspond to an electric susceptibilitythat is positive in a first direction and negative in a second direction(where the first and second directions may correspond to the axial andtransverse directions 150 and 151 in FIGS. 4 and 7, or vice versa). Thenthe propagating electromagnetic wave 125 may induce an electricpolarization in the first direction that positively corresponds (inaccordance with the positive electric susceptibility) to an electricfield component of the propagating electromagnetic wave in the firstdirection, and further induce an electric polarization in the seconddirection that negatively corresponds (in accordance with the negativeelectric susceptibility) to an electric field component of thepropagating electromagnetic wave in the second direction. In analternate example, a conversion structure 100 as in FIGS. 4 and 7 mayprovide an indefinite medium having an indefinite permeability, and theindefinite permeability may correspond to a magnetic susceptibility thatis positive in a first direction and negative in a second direction(where the first and second directions may correspond to the axial andtransverse directions 150 and 151 in FIGS. 4 and 7, or vice versa). Thenthe propagating electromagnetic wave 125 may induce a magneticpolarization in the first direction that positively corresponds (inaccordance with the positive magnetic susceptibility) to a magneticfield component of the propagating electromagnetic wave in the firstdirection, and further induce a magnetic polarization in the seconddirection that negatively corresponds (in accordance with the negativemagnetic susceptibility) to a magnetic field component of thepropagating electromagnetic wave in the second direction. Operation 1320optionally further includes sub-operation 1323—inducing a thirdpolarization in a third direction, the third polarization positivelycorresponding to a third component of the electromagnetic field in thethird direction-and sub-operation 1324—inducing a fourth polarization ina fourth direction perpendicular to the third direction, the fourthpolarization negatively corresponding to a fourth component of theelectromagnetic field in the fourth direction. For example, a conversionstructure 100 as in FIGS. 4 and 7 may provide an indefinite mediumhaving both an indefinite permittivity and an indefinite permeability,the indefinite permittivity corresponding to a electric susceptibilitythat is positive in a first direction and negative in a second direction(where the first and second directions may correspond to the axial andtransverse directions 150 and 151 in FIGS. 4 and 7, or vice versa) andthe indefinite permeability corresponding to a magnetic susceptibilitythat is positive in a third direction and negative in a fourth direction(where the third and fourth directions may correspond to the axial andtransverse directions 150 and 151 of FIGS. 4 and 7, or vice versa). Thenthe propagating electromagnetic wave 125 may induce: (1) an electricpolarization in the first direction that positively corresponds (inaccordance with the positive electric susceptibility) to an electricfield component of the propagating electromagnetic wave in the firstdirection, (2) an electric polarization in the second direction thatnegatively corresponds (in accordance with the negative electricsusceptibility) to an electric field component of the propagatingelectromagnetic wave in the second direction, (3) a magneticpolarization in the third direction that positively corresponds (inaccordance with the positive magnetic susceptibility) to a magneticfield component of the propagating electromagnetic wave in the seconddirection, and (4) a magnetic polarization in the fourth direction thatnegatively corresponds (in accordance with the negative magneticsusceptibility) to a magnetic field component of the propagatingelectromagnetic wave in the fourth direction. Flow 1300 includesoperation 1330—coupling the propagated electromagnetic energy to anevanescent electromagnetic wave at the output region. For example, theconversion structure 100 of FIGS. 4 and 7 provides (in a reciprocalscenario to that of FIGS. 4 and 7, as described previously) anevanescent electromagnetic wave 120 at the first surface region 111 (theevanescent wave having an exponential decay away from the conversionstructure for this reciprocal scenario, not decaying towards theconversion structure as depicted).

With reference now to FIG. 14, an illustrative embodiment is depicted asa system block diagram. The system 1400 includes an evanescentconversion unit 1420 optionally coupled to a control unit 1440. Theevanescent conversion unit 1420 may include a conversion structure suchas that depicted as element 100 in FIGS. 1-11. The conversion structuremay be a variable conversion structure, such as a variable metamaterialresponsive to one or more control inputs to vary one or more operatingcharacteristics (operating frequency, operating wave polarization,effective coordinate transformation for a transformation medium, etc),and the control unit 1440 may include control circuitry that providesone or more control inputs to the variable conversion structure. Theevanescent conversion unit 1420 may further include a positioningstructure (e.g. with one or more piezo stages, nanopositioners,conveyors/turntables, or other actuators) having one or more controlinputs to vary a position/orientation of the conversion structure and/orvary a position/orientation of a sample or target in relation to theconversion structure (e.g. within an evanescent range of the conversionstructure), and the control unit 1440 may include control circuitry thatprovides the one or more control inputs to the positioning structure,optionally in response to a feedback signal from the positioningstructure (e.g. a cantilever force feedback). The evanescent conversionunit 1420 may include one or more optical components, e.g. positioned todeliver electromagnetic energy to an input surface of the conversionstructure, receive electromagnetic energy from an output surface of theconversion structure, deliver electromagnetic energy to a sample ortarget positioned within an evanescent range of the conversionstructure, and/or receive electromagnetic energy from a sample or targetpositioned within an evanescent range of the conversion structure; andthe control unit 1440 may include control circuitry that provides one ormore control inputs to the one or more optical components (e.g. tocontrol orientations, focusing characteristics, aperture sizes, etc.).The system optionally includes an input unit 1410 coupled to theevanescent conversion unit 1420 (e.g. to deliver electromagnetic energyto the evanescent conversion unit 1420); the input unit may include anelectromagnetic source (e.g. an antenna, laser, or transducer) as wellas input circuitry and/or optical components such as modulators, phaseadjusters, etc. The system optionally includes an output unit 1430coupled to the evanescent conversion unit 1420 (e.g. to receiveelectromagnetic energy from the evanescent conversion unit 1420); theoutput unit may include an electromagnetic detector (e.g. a CCD array,photomultiplier, etc.) as well as output circuitry and/or opticalcomponents such as demodulators, phase adjusters, spectral analyzers,image processing circuitry, etc.

All of the above U.S. patents, U.S. patent application publications,U.S. patent applications, foreign patents, foreign patent applicationsand non-patent publications referred to in this specification and/orlisted in any Application Data Sheet, are incorporated herein byreference, to the extent not inconsistent herewith.

One skilled in the art will recognize that the herein describedcomponents (e.g., steps), devices, and objects and the discussionaccompanying them are used as examples for the sake of conceptualclarity and that various configuration modifications are within theskill of those in the art. Consequently, as used herein, the specificexemplars set forth and the accompanying discussion are intended to berepresentative of their more general classes. In general, use of anyspecific exemplar herein is also intended to be representative of itsclass, and the non-inclusion of such specific components (e.g., steps),devices, and objects herein should not be taken as indicating thatlimitation is desired.

With respect to the use of substantially any plural and/or singularterms herein, those having skill in the art can translate from theplural to the singular and/or from the singular to the plural as isappropriate to the context and/or application. The varioussingular/plural permutations are not expressly set forth herein for sakeof clarity.

While particular aspects of the present subject matter described hereinhave been shown and described, it will be apparent to those skilled inthe art that, based upon the teachings herein, changes and modificationsmay be made without departing from the subject matter described hereinand its broader aspects and, therefore, the appended claims are toencompass within their scope all such changes and modifications as arewithin the true spirit and scope of the subject matter described herein.Furthermore, it is to be understood that the invention is defined by theappended claims. It will be understood by those within the art that, ingeneral, terms used herein, and especially in the appended claims (e.g.,bodies of the appended claims) are generally intended as “open” terms(e.g., the term “including” should be interpreted as “including but notlimited to,” the term “having” should be interpreted as “having atleast,” the term “includes” should be interpreted as “includes but isnot limited to,” etc.). It will be further understood by those withinthe art that if a specific number of an introduced claim recitation isintended, such an intent will be explicitly recited in the claim, and inthe absence of such recitation no such intent is present. For example,as an aid to understanding, the following appended claims may containusage of the introductory phrases “at least one” and “one or more” tointroduce claim recitations. However, the use of such phrases should notbe construed to imply that the introduction of a claim recitation by theindefinite articles “a” or “an” limits any particular claim containingsuch introduced claim recitation to inventions containing only one suchrecitation, even when the same claim includes the introductory phrases“one or more” or “at least one” and indefinite articles such as “a” or“an” (e.g., “a” and/or “an” should typically be interpreted to mean “atleast one” or “one or more”); the same holds true for the use ofdefinite articles used to introduce claim recitations. In addition, evenif a specific number of an introduced claim recitation is explicitlyrecited, those skilled in the art will recognize that such recitationshould typically be interpreted to mean at least the recited number(e.g., the bare recitation of “two recitations,” without othermodifiers, typically means at least two recitations, or two or morerecitations). Furthermore, in those instances where a conventionanalogous to “at least one of A, B, and C, etc.” is used, in generalsuch a construction is intended in the sense one having skill in the artwould understand the convention (e.g., “a system having at least one ofA, B, and C” would include but not be limited to systems that have Aalone, B alone, C alone, A and B together, A and C together, B and Ctogether, and/or A, B, and C together, etc.). In those instances where aconvention analogous to “at least one of A, B, or C, etc.” is used, ingeneral such a construction is intended in the sense one having skill inthe art would understand the convention (e.g., “a system having at leastone of A, B, or C” would include but not be limited to systems that haveA alone, B alone, C alone, A and B together, A and C together, B and Ctogether, and/or A, B, and C together, etc.). It will be furtherunderstood by those within the art that virtually any disjunctive wordand/or phrase presenting two or more alternative terms, whether in thedescription, claims, or drawings, should be understood to contemplatethe possibilities of including one of the terms, either of the terms, orboth terms. For example, the phrase “A or B” will be understood toinclude the possibilities of “A” or “B” or “A and B.”

With respect to the appended claims, those skilled in the art willappreciate that recited operations therein may generally be performed inany order. Examples of such alternate orderings may include overlapping,interleaved, interrupted, reordered, incremental, preparatory,supplemental, simultaneous, reverse, or other variant orderings, unlesscontext dictates otherwise. With respect to context, even terms like“responsive to,” “related to,” or other past-tense adjectives aregenerally not intended to exclude such variants, unless context dictatesotherwise.

While various aspects and embodiments have been disclosed herein, otheraspects and embodiments will be apparent to those skilled in the art.The various aspects and embodiments disclosed herein are for purposes ofillustration and are not intended to be limiting, with the true scopeand spirit being indicated by the following claims.

1-154. (canceled)
 155. A method, comprising: coupling to an evanescentelectromagnetic wave at an input region; responsive to the coupling,propagating electromagnetic energy in an electromagnetic field from afirst surface within the input region to a second surface within anoutput region, where one of the first and second surfaces is asubstantially planar surface and the other of the first and secondsurfaces is a substantially nonplanar surface, and where the propagatingincludes: inducing a first polarization in a first direction, the firstpolarization positively corresponding to a first component of theelectromagnetic field in the first direction; and inducing a secondpolarization in a second direction perpendicular to the first direction,the second polarization negatively corresponding to a second componentof the electromagnetic field in the second direction; and providing thepropagated electromagnetic energy as a non-evanescent electromagneticwave at the output region.
 156. (canceled)
 157. The method of claim 155,wherein the first surface is an exterior surface of the input region.158. (canceled)
 159. The method of claim 155, wherein the second surfaceis an exterior surface portion of the output region.
 160. The method ofclaim 155, wherein the first surface is the substantially planarsurface.
 161. The method of claim 155, wherein the second surface is thesubstantially planar surface.
 162. The method of claim 155, wherein thesubstantially nonplanar surface is substantially concave towards thesubstantially planar surface.
 163. The method of claim 155, wherein thesubstantially nonplanar surface is substantially convex towards thesubstantially planar surface.
 164. (canceled)
 165. The method of claim155, wherein the first polarization is a first electric polarization,the first component is a first electric component, the secondpolarization is a second electric polarization, and the second componentis a second electric component.
 166. The method of claim 155, whereinthe first polarization is a first magnetic polarization, the firstcomponent is a first magnetic component, the second polarization is asecond magnetic polarization, and the second component is a secondmagnetic component.
 167. (canceled)
 168. The method of claim 155,wherein the first direction is a non-constant direction that is afunction of location intermediate the first and second surfaces. 169.(canceled)
 170. The method of claim 155, wherein the second direction isa non-constant direction that is a function of location intermediate thefirst and second surfaces.
 171. The method of claim 155, wherein thepropagating further includes: inducing a third polarization in a thirddirection, the third polarization positively corresponding to a thirdcomponent of the electromagnetic field in the third direction; andinducing a fourth polarization in a fourth direction perpendicular tothe third direction, the fourth polarization negatively corresponding toa fourth component of the electromagnetic field in the fourth direction.172. The method of claim 171, wherein the third direction issubstantially equal to the first direction.
 173. The method of claim171, wherein the third direction is substantially equal to the seconddirection.
 174. The method of claim 171, wherein the third direction issubstantially mutually perpendicular to the first and second directions.175. The method of claim 171, wherein the fourth direction issubstantially equal to the first direction.
 176. The method of claim171, wherein the fourth direction is substantially equal to the seconddirection.
 177. The method of claim 171, wherein the fourth direction issubstantially mutually perpendicular to the first and second directions.178. The method of claim 171, wherein the first polarization is a firstelectric polarization, the first component is a first electriccomponent, the second polarization is a second electric polarization,the second component is a second electric component, the thirdpolarization is a first magnetic polarization, the third component is afirst magnetic component, the fourth polarization is a second magneticpolarization, and the fourth component is a second magnetic component.179. The method of claim 155, wherein the propagating is a propagatingalong a propagation direction that substantially coincides with thefirst direction.
 180. The method of claim 155, wherein the propagatingis a propagating along a propagation direction that substantiallycoincides with the second direction.
 181. The method of claim 155,wherein the propagating is a propagating along at least two propagationdirections, each of the at least two propagation directionssubstantially having a common angle with respect to the first direction.182. The method of claim 155, wherein the propagating is a propagatingalong at least two propagation directions, each of the at least twopropagation directions substantially having a common angle with respectto the second direction. 183-190. (canceled)
 191. The method of claim155, wherein the coupling to the evanescent electromagnetic wave is acoupling to the evanescent electromagnetic wave at a first frequencywith a first wavenumber, and the providing of the non-evanescentelectromagnetic wave is a providing of the non-evanescentelectromagnetic wave at the first frequency with a second wavenumber,the first wavenumber corresponding to a surface parallel direction ofthe first surface and the second wavenumber corresponding to a surfaceparallel direction of the second surface.
 192. The method of claim 191,wherein the first and second surfaces define an interior regionintermediate the first and second surfaces, a first exterior regionoutside the interior region and adjacent to the first surface defines afirst phase velocity for electromagnetic radiation, and the firstwavenumber is greater than the first frequency divided by the firstphase velocity.
 193. (canceled)
 194. The method of claim 192, wherein asecond exterior region outside the interior region and adjacent to thesecond surface defines a second phase velocity for electromagneticradiation, and the second wavenumber is less than the first frequencydivided by the second phase velocity.
 195. (canceled)
 196. (canceled)197. (canceled)
 198. The method of claim 192, where the electromagneticfield defines a transverse wavenumber, and the propagating ofelectromagnetic energy in the electromagnetic field within the interiorregion provides a substantially continuous variation of the transversewavenumber from the first wavenumber at the first surface to the secondwavenumber at the second surface. 199-205. (canceled)
 206. A method,comprising: receiving a non-evanescent electromagnetic wave at an inputregion; responsive to the receiving, propagating electromagnetic energyin an electromagnetic field from a first surface within the input regionto a second surface within an output region, where one of the first andsecond surfaces is a substantially planar surface and the other of thefirst and second surfaces is a substantially nonplanar surface, andwhere the propagating includes: inducing a first polarization in a firstdirection, the first polarization positively corresponding to a firstcomponent of the electromagnetic field in the first direction; andinducing a second polarization in a second direction perpendicular tothe first direction, the second polarization negatively corresponding toa second component of the electromagnetic field in the second direction;and coupling the propagated electromagnetic energy to an evanescentelectromagnetic wave at the output region.
 207. (canceled)
 208. Themethod of claim 155, wherein the first surface is an exterior surface ofthe input region.
 209. (canceled)
 210. The method of claim 155, whereinthe second surface is an exterior surface portion of the output region.211. The method of claim 206, wherein the first surface is thesubstantially planar surface.
 212. The method of claim 206, wherein thesecond surface is the substantially planar surface.
 213. The method ofclaim 206, wherein the substantially nonplanar surface is substantiallyconcave towards the substantially planar surface.
 214. The method ofclaim 206, wherein the substantially nonplanar surface is substantiallyconvex towards the substantially planar surface.
 215. (canceled) 216.The method of claim 206, wherein the first polarization is a firstelectric polarization, the first component is a first electriccomponent, the second polarization is a second electric polarization,and the second component is a second electric component.
 217. The methodof claim 206, wherein the first polarization is a first magneticpolarization, the first component is a first magnetic component, the218. (canceled)
 219. The method of claim 206, wherein the firstdirection is a non-constant direction that is a function of locationintermediate the first and second surfaces.
 220. (canceled)
 221. Themethod of claim 206, wherein the second direction is a non-constantdirection that is a function of location intermediate the first andsecond surfaces.
 222. The method of claim 206, wherein the propagatingfurther includes: inducing a third polarization in a third direction,the third polarization positively corresponding to a third component ofthe electromagnetic field in the third direction; and inducing a fourthpolarization in a fourth direction perpendicular to the third direction,the fourth polarization negatively corresponding to a fourth componentof the electromagnetic field in the fourth direction.
 223. The method ofclaim 222, wherein the third direction is substantially equal to thefirst direction.
 224. The method of claim 222, wherein the thirddirection is substantially equal to the second direction.
 225. Themethod of claim 222, wherein the third direction is substantiallymutually perpendicular to the first and second directions.
 226. Themethod of claim 222, wherein the fourth direction is substantially equalto the first direction.
 227. The method of claim 222, wherein the fourthdirection is substantially equal to the second direction.
 228. Themethod of claim 222, wherein the fourth direction is substantiallymutually perpendicular to the first and second directions.
 229. Themethod of claim 222, wherein the first polarization is a first electricpolarization, the first component is a first electric component, thesecond polarization is a second electric polarization, the secondcomponent is a second electric component, the third polarization is afirst magnetic polarization, the third component is a first magneticcomponent, the fourth polarization is a second magnetic polarization,and the fourth component is a second magnetic component.
 230. The methodof claim 206, wherein the propagating is a propagating along apropagation direction that substantially coincides with the firstdirection.
 231. The method of claim 206, wherein the propagating is apropagating along a propagation direction that substantially coincideswith the second direction.
 232. The method of claim 206, wherein thepropagating is a propagating along at least two propagation directions,each of the at least two propagation directions substantially having acommon angle with respect to the first direction.
 233. The method ofclaim 206, wherein the propagating is a propagating along at least twopropagation directions, each of the at least two propagation directionssubstantially having a common angle with respect to the seconddirection. 234-240. (canceled)
 241. The method of claim 206, wherein thereceiving of the non-evanescent electromagnetic wave is a receiving ofthe non-evanescent electromagnetic wave at a first frequency with afirst wavenumber, and the coupling to the evanescent electromagneticwave is a coupling to the evanescent electromagnetic wave at the firstfrequency with a second wavenumber, the first wavenumber wave is acoupling to the evanescent electromagnetic wave at the first frequencywith a second wavenumber, the first wavenumber corresponding to asurface parallel direction of the first surface and the secondwavenumber corresponding to a surface parallel direction of the secondsurface.
 242. The method of claim 241, wherein the first and secondsurfaces define an interior region intermediate the first and secondsurfaces, a first exterior region outside the interior region andadjacent to the first surface defines a first phase velocity forelectromagnetic radiation, and the first wavenumber is less than thefirst frequency divided by the first phase velocity.
 243. (canceled)244. The method of claim 242, wherein a second exterior region outsidethe interior region and adjacent to the second surface defines a secondphase velocity for electromagnetic radiation, and the second wavenumberis greater than the first frequency divided by the second phasevelocity.
 245. (canceled)
 246. (canceled)
 247. (canceled)
 248. Themethod of claim 242, where the electromagnetic field defines atransverse wavenumber, and the propagating of electromagnetic energy inthe electromagnetic field within the interior region provides asubstantially continuous variation of the transverse wavenumber from thefirst wavenumber at the first surface to the second wavenumber at thesecond surface. 249-255. (canceled)